Laplace Transform

Contents

**Describe Properties of Laplace Transform** 2

**: Evaluate Laplace Transform of Elementary Functions** 3

**Describe Scale Change Property** 5

**Evaluate Laplace Transform using Scale Change Property** 6

**Describe First Shifting Property** 8

**Evaluate Laplace Transform using First Shifting Property** 10

**Evaluate Laplace Transform of a Function Multiplied by t ^{n}** 12

**Evaluate Laplace Transform of a Function Divided by t** 14

**Describe Unit-Step Function** 15

**Evaluate Laplace Transform of a Unit-step Function** 15

**Describe Second Shifting Property** 15

**Describe Dirac-Delta(Impulse) Function** 15

**Evaluate Laplace Transform of a Dirac-Delta (Impulse) Function** 15

**Evaluate Laplace Transform of Periodic Function** 15

**Evaluate Laplace Transform of Error Function** 15

**Describe Inverse Laplace Transform** 15

**Evaluate the Inverse Laplace Transform of Elementary Functions** 15

**Evaluate the Inverse Laplace Transform of Functions Multiplied by s** 15

**Evaluate the Inverse Laplace Transform of functions Divided by s** 15

**Evaluate the Inverse Laplace Transform of functions by using Partial Fraction Method** 15

**Evaluate the Inverse Laplace Transform of functions by using Convolution Theorem** 15

**Evaluate Laplace Transform of Derivatives and Integrals** 15

**Evaluate Inverse Laplace Transform of Derivatives and Integrals** 15

**Find the solution of Linear Differential Equations by using Laplace Transform** 15

**Find the solution of Simultaneous Differential Equations by using Laplace Transform** 15

**Describe Laplace Transform**

The Laplace transform is a powerful mathematical technique that converts a function of time into a function of complex frequency. This transformation is used extensively in engineering, physics, and other fields to solve differential equations and perform other mathematical analyses.

The Laplace transform of a function f(t) is denoted by F(s) and is defined by the following integral:

F(s) = L{f(t)} = e^{(-st)} f(t) dt

where s is a complex frequency parameter. The Laplace transform essentially converts a function from the time domain to the s-domain, where it can be analyzed more easily.

The Laplace transform has several important properties that make it a powerful tool for solving differential equations and other problems. For example, it is a linear operator, which means that if a linear combination of functions is transformed, the result is the same linear combination of the transformed functions. Additionally, the Laplace transform has a shifting property that allows the time domain function to be shifted in time by a constant and the frequency domain function to be shifted in frequency by the same constant.

The Laplace transform is often used to solve initial value problems, where the value of a function and its derivative at a specific time are given. By applying the Laplace transform to the differential equation that describes the system, it can be solved in the frequency domain and then transformed back into the time domain to obtain the solution. The Laplace transform is also useful for solving integral equations and for analyzing control systems, circuits, and other systems with complex dynamics.

An example of Laplace transform in action is the analysis of an electrical circuit. By applying the Laplace transform to the circuit’s governing differential equations, one can obtain a transfer function that describes the relationship between the input and output signals of the circuit. This transfer function can be used to design filters, amplifiers, and other signal processing circuits.

**Describe Properties of Laplace Transform**

The Laplace transform is a mathematical tool that is used to transform a function of time into a function of a complex variable s, which is known as the Laplace variable. The Laplace transform has many properties that make it a useful tool in solving differential equations and analyzing linear time-invariant systems. Some of the most important properties of the Laplace transform are:

- Linearity: The Laplace transform is a linear operator, which means that it satisfies the following property:

L{a*f(t) + b*g(t)} = a*L{f(t)} + b*L{g(t)}

where a and b are constants, and f(t) and g(t) are functions of time. - Time-Shifting: The Laplace transform of a time-shifted function is related to the Laplace transform of the original function by a simple exponential factor:

L{f(t – a)} = e^{(-as)}* L{f(t)}

This property is useful for finding the Laplace transform of functions that are shifted in time. - Frequency-Shifting: The Laplace transform of a function that has been multiplied by a complex exponential is related to the Laplace transform of the original function by a simple frequency-shift:

L{e^{(at)}* f(t)} = L{f(t – a)}

This property is useful for finding the Laplace transform of functions that are modulated by a complex exponential. - Differentiation in Time: The Laplace transform of the derivative of a function f(t) is related to the Laplace transform of the original function by a simple algebraic expression:

L{f'(t)} = s * L{f(t)} – f(0)

where f(0) is the initial value of the function. - Integration in Time: The Laplace transform of the integral of a function f(t) is related to the Laplace transform of the original function by a simple algebraic expression:

L{ f(τ) dτ} = 1/s * L{f(t)} - Convolution: The Laplace transform of the convolution of two functions f(t) and g(t) is related to the product of their individual Laplace transforms:

L{f(t) * g(t)} = L{f(t)} * L{g(t)}

where * denotes convolution and L{} denotes the Laplace transform.

These properties, along with a few others, make the Laplace transform a powerful tool for solving differential equations and analyzing linear systems. By applying these properties to the Laplace transform of a given function, we can often simplify the problem and find a solution more easily.

**: Evaluate Laplace Transform of Elementary Functions**

The Laplace transform is a mathematical technique that can be used to solve differential equations. It involves transforming a time-domain function into a frequency-domain function. This transformation can be useful for solving differential equations because it can turn a differential equation into an algebraic equation.

The Laplace transform of a function f(t) is denoted by L{f(t)} and is defined by the integral:

L{f(t)} = F(s) = f(t)e^{(-st)} dt

where s is a complex variable. The Laplace transform is a linear operator, which means that L{af(t) + bg(t)} = aL{f(t)} + bL{g(t)}.

The Laplace transform of some elementary functions are:

- L{1} = 1/s
- L{t} = 1/s
^{2} - L{t
^{n}} = n!/s^{(n+1)}, where n is a non-negative integer - L{e
^{(at)}} = 1/(s-a), where a is a constant - L{sin(bt)} = b/(s
^{2}+ b^{2}) - L{cos(bt)} = s/(s
^{2}+ b^{2}) - L{sinh(at)} = a/(s
^{2}– a^{2}) - L{cosh(at)} = s/(s
^{2 }– a^{2})

For example, to find the Laplace transform of the function f(t) = 2e^{(-3t)}, we can use the definition of the Laplace transform:

L{f(t)} = 2e^{(-3t)} e^{(-st) }dt

= 2/(s+3)

Similarly, the Laplace transform of the function f(t) = sin(2t) is:

L{f(t)} = sin(2t) e^{(-st) }dt

= 2/(s^{2} + 4)

The Laplace transform can also be used to solve initial value problems involving differential equations. For example, consider the differential equation:

y”(t) + 4y'(t) + 3y(t) = 0, y(0) = 1, y'(0) = 0

Taking the Laplace transform of both sides of the equation, we get:

s^{2} Y(s) – s + 4s Y(s) – 4 + 3Y(s) = 0

Solving for Y(s), we get:

Y(s) = 1/(s^{2} + 4s + 3)

Using partial fraction decomposition, we can write:

Y(s) = 1/((s+1)(s+3))

Taking the inverse Laplace transform, we get:

y(t) = e^{(-t)} – e^{(-3t)}

Thus, we have solved the initial value problem using Laplace transform.

**Describe Scale Change Property**

The scale change property of the Laplace transform states that if a function f(t) is scaled horizontally by a factor of ‘a’, then its Laplace transform F(s) is scaled vertically by a factor of 1/a. This means that if f(at) has the Laplace transform F(s), then f(t) has the Laplace transform (1/a)F(s/a).

Mathematically, the scale change property can be expressed as follows:

L{f(at)} = e^{(-st)} f(at) dt = (1/a) e^{(-s/a)}(f(t) dt) = (1/a) F(s/a)

where L denotes the Laplace transform operator.

This property is useful in solving differential equations involving functions that are scaled or compressed in time. For example, if a function f(t) is defined as f(t) = e^{(-2t)}, then its Laplace transform is F(s) = 1/(s+2). If we want to find the Laplace transform of g(t) = f(2t), we can use the scale change property to get:

L{g(t)} = L{f(2t)} = (1/2) F(s/2) = 1/(2s+4)

Similarly, if we want to find the Laplace transform of h(t) = f(3t), we can use the scale change property to get:

L{h(t)} = L{f(3t)} = (1/3) F(s/3) = 1/(3s+6)

Thus, the scale change property simplifies the Laplace transform computation for functions that are scaled horizontally.

**Evaluate Laplace Transform using Scale Change Property**

The Laplace transform is a mathematical technique used to solve differential equations by converting them into algebraic equations. It is defined as the integral of a function multiplied by an exponential function, where the exponential function has a complex exponent. The Laplace transform is used in many areas of science and engineering, including control systems, signal processing, and communication systems. One of the fundamental properties of the Laplace transform is the Scale Change Property, which allows us to change the scaling of a function in the time domain to a scaling in the Laplace domain.

Scale Change Property of Laplace Transform:

If a function f(t) is transformed into F(s) using Laplace transform, then scaling the function by a constant α in the time domain results in a scaling of F(s) by 1/α in the Laplace domain.

Mathematically, the Scale Change Property can be written as:

L{f(αt)} = (1/α)F(s/α)

where L{f(t)} = F(s) is the Laplace transform of f(t).

Example 1: Find the Laplace transform of f(t) = e^{(2t)}.

Solution:

Using the Laplace transform definition, we have:

L{e^{(2t)}} = e^{(-st)} e^{(2t)} dt

= e^{(t(2-s))} dt

= 1/(s-2)

Now, using the Scale Change Property, we can find the Laplace transform of f(2t):

L{e^{(4t)}} = (1/2) L{e^{(2t)}}

= (1/2) (1/(s-2))

= 1/(2(s-2))

Example 2: Find the Laplace transform of f(t) = sin(3t).

Solution:

Using the Laplace transform definition, we have:

L{sin(3t)} = e^{(-st)} sin(3t) dt

= (1/2i) [e^{((3i-s)t)} – e^{(-(3i+s)t)}] dt

= (3/(s^{2} + 9))

Now, using the Scale Change Property, we can find the Laplace transform of f(2t):

L{sin(6t)} = (1/2) L{sin(3t)}

= (1/2) (3/(s^{2} + 9))

= (3/2(s^{2} + 9))

In summary, the Scale Change Property of Laplace Transform is a useful tool to evaluate the Laplace transform of a function that is scaled in the time domain. By using this property, we can relate the Laplace transform of a scaled function to the Laplace transform of the original function. This property is often used in the analysis and design of systems where scaling plays an important role.

**Describe First Shifting Property**

The Laplace transform is a powerful mathematical tool that is widely used in many fields of science and engineering. It is a method of converting a function of time into a function of complex frequency. The Laplace transform has many useful properties, which enable us to manipulate and analyze functions in the Laplace domain. One of these properties is the First Shifting Property, which enables us to shift a function in the time domain by a constant amount and determine its Laplace transform.

First Shifting Property of Laplace Transform:

If a function f(t) is transformed into F(s) using Laplace transform, then shifting the function by a constant a in the time domain results in multiplying F(s) by e^{(-as)}.

Mathematically, the First Shifting Property can be written as:

L{f(t-a)} = e^{(-as)} F(s)

where L{f(t)} = F(s) is the Laplace transform of f(t).

Example 1: Find the Laplace transform of f(t) = e^{(2t)} for t ≥ 0.

Solution:

Using the Laplace transform definition, we have:

L{e^{(2t)}} = e^{(-st)} e^{(2t)} dt

= e^{((2-s)t)} dt

= (1/(s-2))

Now, using the First Shifting Property, we can find the Laplace transform of f(t-3):

L{e^{(2(t-3))}} = e^{(-3s)} L{e^{(2t)}}

= (1/(s-2)) e^{(-3s)}

Example 2: Find the Laplace transform of f(t) = sin(t) for t ≥ 0.

Solution:

Using the Laplace transform definition, we have:

L{sin(t)} = e^{(-st)} sin(t) dt

= (1/(s^{2} + 1))

Now, using the First Shifting Property, we can find the Laplace transform of f(t-π/2):

L{sin(t-π/2)} = e^{(-πs/2)} L{sin(t)}

= (1/(s^{2} + 1)) e^{(-πs/2)}

In summary, the First Shifting Property of Laplace Transform is a useful tool to evaluate the Laplace transform of a function that is shifted in the time domain. By using this property, we can relate the Laplace transform of a shifted function to the Laplace transform of the original function. This property is often used in the analysis and design of systems where shifting plays an important role.

**Evaluate Laplace Transform using First Shifting Property**

The Laplace transform is a mathematical tool used in many fields of science and engineering to transform a function from the time domain to the frequency domain. One of the most useful properties of the Laplace transform is the First Shifting Property. This property enables us to shift a function in the time domain by a constant amount and determine its Laplace transform. In this learning outcome, we will learn how to evaluate the Laplace transform using the First Shifting Property.

First Shifting Property of Laplace Transform:

If a function f(t) is transformed into F(s) using Laplace transform, then shifting the function by a constant a in the time domain results in multiplying F(s) by e^{(-as)}.

Mathematically, the First Shifting Property can be written as:

L{f(t-a)} = e^{(-as)} F(s)

where L{f(t)} = F(s) is the Laplace transform of f(t).

Example 1: Find the Laplace transform of f(t) = 3e^{(4t)} for t ≥ 0.

Solution:

Using the Laplace transform definition, we have:

L{3e^{(4t)}} = 3/(s-4)

Now, using the First Shifting Property, we can find the Laplace transform of f(t-2):

L{3e^{(4(t-2))}} = e^{(-8s)} L{3e^{(4t)}}

= 3/(s-4) e^{(-8s)}

Example 2: Find the Laplace transform of f(t) = 2sin(3t) for t ≥ 0.

Solution:

Using the Laplace transform definition, we have:

L{2sin(3t)} = 6/(s^{2} + 9)

Now, using the First Shifting Property, we can find the Laplace transform of f(t-π/3):

L{2sin(3(t-π/3))} = e^{(-πs/3)} L{2sin(3t)}

= 6/(s^{2} + 9) e^{(-πs/3)}

Example 3: Find the Laplace transform of f(t) = e^{(-2t)} cos(4t) for t ≥ 0.

Solution:

Using the Laplace transform definition, we have:

L{e^{(-2t)} cos(4t)} = s+2/(s+2)^{2} + 16

Now, using the First Shifting Property, we can find the Laplace transform of f(t-π/4):

L{e^{(-2(t-π/4)) }cos(4(t-π/4))} = e^{(πs/4)} L{e^{(-2t)} cos(4t)}

= (s+2)/(s+2)^{2} + 16 e^{(πs/4)}

In summary, the First Shifting Property of Laplace Transform is a powerful tool that enables us to relate the Laplace transform of a shifted function to the Laplace transform of the original function. By using this property, we can evaluate the Laplace transform of a function that is shifted in the time domain. This property is often used in the analysis and design of systems where shifting plays an important role.

**Evaluate Laplace Transform of a Function Multiplied by t**^{n}

^{n}

In the study of Laplace transform, it is important to be able to evaluate the transform of a function multiplied by a power of t. This is important in the study of differential equations, as many differential equations involve t^{n} multiplied by a function. In this learning outcome, we will learn how to evaluate the Laplace transform of a function multiplied by t^{n}.

The Laplace transform of a function f(t) is defined as:

F(s) = L{f(t)} = e^{(-st)} f(t) dt

If f(t) is multiplied by t^{n}, we can evaluate its Laplace transform using the following formula:

L{t^{n} f(t)} = (-1)^{n} F^{(n)}(s)

where F^{(n)}(s) denotes the nth derivative of F(s) with respect to s.

Example 1: Find the Laplace transform of t^{2} e^{(-3t)}.

Solution:

Using the Laplace transform definition, we have:

L{t^{2} e^{(-3t)}} = e^{(-st)} t^{2} e^{(-3t)} dt

We can rewrite this integral as:

L{t^{2} e^{(-3t)}} = t^{2} e^{(-(s+3)t)} dt

To evaluate this integral, we need to use integration by parts twice:

L{t^{2} e^{(-3t)}} =_{0}^{∞} + 2/s+3 ∫[0, ∞) t e^{(-(s+3)t)} dt

L{t^{2} e^{(-3t)}} = 2 / (s+3)^{2}

Using the formula L{t^{n} f(t)} = (-1)^{n} F^{(n})(s), we have:

L{t^{2} e^{(-3t)}} = (-1)^{2} F”(s) = 2 / (s+3)^{2}

Example 2: Find the Laplace transform of t cos(t).

Solution:

Using the Laplace transform definition, we have:

L{t cos(t)} = e^{(-st) }t cos(t) dt

We can use integration by parts to evaluate this integral:

L{t cos(t)} = [t sin(t) / s]_{0}^{∞} – sin(t) / s dt

L{t cos(t)} = – 1 / s sin(t) dt = -1 / (s^{2} + 1)

Using the formula L{t^{n} f(t)} = (-1)^{n} F^{(n)}(s), we have:

L{t cos(t)} = (-1)^{1} F'(s) = -s / (s^{2 }+ 1)

**Evaluate Laplace Transform of a Function Divided by t**

In Laplace transform, it is important to be able to evaluate the transform of a function divided by t. This situation arises in many practical problems, such as circuits with a resistor and capacitor in parallel, where the voltage across the capacitor is proportional to the charge on the capacitor, which is the integral of the current. In this learning outcome, we will learn how to evaluate the Laplace transform of a function divided by t.

The Laplace transform of a function f(t) is defined as:

F(s) = L{f(t)} = e^{(-st)} f(t) dt

If f(t) is divided by t, we can evaluate its Laplace transform using the following formula:

L{f(t) / t} = F(σ) dσ

where F(σ) is the Laplace transform of f(t).

Example 1: Find the Laplace transform of sin(t) / t.

Solution:

Using the formula L{f(t) / t} = F(σ) dσ, we have:

L{sin(t) / t} = L{sin(t)} dσ

Using the Laplace transform of sin(t), we have:

L{sin(t)} = 1 / (s^{2} + 1)

Therefore, we have:

L{sin(t) / t} = 1 / (σ^{2} + 1) dσ

To evaluate this integral, we can use the substitution σ = tan(u), which gives:

L{sin(t) / t} = du = π/2 – arctan(s)

Example 2: Find the Laplace transform of ln(t) / t.

Solution:

Using the formula L{f(t) / t} = F(σ) dσ, we have:

L{ln(t) / t} = L{ln(t)} dσ

Using the Laplace transform of ln(t), we have:

L{ln(t)} = -γ – ln(s)

where γ is the Euler-Mascheroni constant.

Therefore, we have:

L{ln(t) / t} = (-γ – ln(σ)) dσ

L{ln(t) / t} = -γ (σ – s) – (σ ln(σ) – σ + s ln(s) – s)

Example 3: Find the Laplace transform of t / (t^{2} + 1).

Solution:

Using the formula L{f(t) / t} = F(σ) dσ, we have:

L{t / (t^{2} + 1)} = L{t^{2} + 1} dσ

Using the Laplace transform of t^{2} + 1, we have:

L{t^{2} + 1} = 2 / s^{3} + s

Therefore, we have:

L{t / (t^{2} + 1)} = 2 / (σ^{3} + σ) dσ

**Describe Unit-Step Function**

In mathematics and engineering, the unit step function, also known as the Heaviside step function, is a function that has a value of 0 for negative input and a value of 1 for non-negative input. It is defined as:

u(t) = { 0, t < 0

{ 1, t >= 0

The unit step function is a common function used in Laplace and Fourier transforms to model physical systems. It is particularly useful in modelling systems that turn on or off at a particular time. The unit step function can also be used to represent a signal that has a value of 0 before a certain time and a value of 1 after that time.

The unit step function has several important properties, including:

- Shift property: u(t – a) is a shifted version of u(t) to the right by a units.
- Scaling property: u(at) is a scaled version of u(t) with a scale factor of 1/a.
- Differentiation property: The derivative of u(t) is the Dirac delta function, which is defined as the derivative of the unit step function.

Example 1: Draw the graph of the unit step function.

Solution:

The graph of the unit step function is a step function that starts at 0 and jumps to 1 at t=0. The graph can be represented as a straight line with a slope of infinity at t=0.

Example 2: Evaluate the integral t u(t-1) dt.

Solution:

We can use the shift property of the unit step function to rewrite u(t-1) as u(t) shifted to the right by 1 unit. Therefore, we have:

t u(t-1) dt = t u(t) dt

Since u(t) is 0 for t<0 and 1 for t>=0, we can evaluate the integral as:

t u(t) dt = t u(t) – t u(t) dt

The integral t u(t) is equal to 1/2, since t u(t) is 0 for t<0 and t for t>=0. The integral t u(t) is equal to 0, since t u(t) is 0 for t<0 and t for 0<=t<1.

Therefore, we have:

t u(t-1) dt = 1/2 – 0 = 1/2.

**Evaluate Laplace Transform of a Unit-step Function**

The Laplace Transform is a mathematical technique used to convert a function of time into a function of a complex variable, usually denoted by ‘s’. The Laplace transform is widely used in electrical engineering, control systems engineering, and signal processing. One of the common functions used in Laplace Transform analysis is the Unit Step Function, also known as the Heaviside step function. The unit step function is defined as follows:

u(t) = 1, t >= 0

u(t) = 0, t < 0

The Laplace Transform of a Unit Step Function is an essential concept to understand for Laplace Transform analysis. The Laplace Transform of the unit step function is given by the following formula:

L{u(t)} = 1/s

Here are some detailed notes for evaluating the Laplace Transform of a Unit-step Function:

- Definition of Laplace Transform: The Laplace Transform of a function f(t) is defined as the integral of the product of f(t) and the exponential function e
^{(-st)}with respect to time ‘t’ from 0 to infinity. This is expressed as follows:

L{f(t)} = f(t) e^{(-st)} dt

- Laplace Transform of the Unit Step Function: To evaluate the Laplace Transform of the unit step function u(t), we use the definition of the Laplace Transform as follows:

L{u(t)} = u(t) e^{(-st)} dt

Since the value of u(t) is equal to 1 for t >= 0, we can change the limits of integration to start from 0, as follows:

L{u(t)} = e^{(-st}) dt

Integrating the above expression yields:

L{u(t)} = [-1/s e^{(-st)}]^{∞}_{0}

Using the limit as t approaches infinity, the exponential term goes to zero, and we are left with:

L{u(t)} = -[1/s e^{(-s∞)}] – (-1/s e^{(-s0)})

The term e^{(-s∞)} goes to zero, so we are left with:

L{u(t)} = 1/s

Therefore, the Laplace Transform of the unit step function u(t) is 1/s.

Example 1: Find the Laplace Transform of the function f(t) = u(t-3)

Solution:

We know that the Laplace Transform of u(t) is 1/s. Therefore, the Laplace Transform of u(t-3) is given by:

L{u(t-3)} = e^{(-3s)} L{u(t)}

Substituting L{u(t)} = 1/s, we get:

L{u(t-3)} = e^{(-3s) }/ s

Example 2: Find the Laplace Transform of the function f(t) = tu(t)

Solution:

We can use the definition of the Laplace Transform to evaluate L{tu(t)} as follows:

L{tu(t)} = tu(t) e^{(-st)} dt

Using integration by parts with u = t and dv = e^{(-st)} dt, we get:

L{tu(t)} = [-e^{(-st)} t / s]_{0}^{∞} + (1/s) e^{(-st)} dt

**Describe Second Shifting Property**

In Laplace Transform analysis, the Second Shifting Property is used to find the Laplace Transform of a function that is a time-shifted version of another function. The Second Shifting Property states that if a function f(t) has the Laplace Transform F(s), then the Laplace Transform of the time-shifted function f(t-a) is given by e^{(-as)} F(s).

Here are some detailed notes for understanding and applying the Second Shifting Property:

- Definition of Laplace Transform: The Laplace Transform of a function f(t) is defined as the integral of the product of f(t) and the exponential function e
^{(-st)}with respect to time ‘t’ from 0 to infinity. This is expressed as follows:

L{f(t)} = f(t) e^{(-st)} dt

- Second Shifting Property: The Second Shifting Property states that if a function f(t) has the Laplace Transform F(s), then the Laplace Transform of the time-shifted function f(t-a) is given by:

L{f(t-a)} = e^{(-as)} F(s)

This property is also called the Time-Shifting Property, and it is used to simplify the calculation of Laplace Transforms for time-shifted functions.

- Proof of the Second Shifting Property: To prove the Second Shifting Property, we use the definition of the Laplace Transform and perform a change of variables. Let g(t) = f(t-a). Then we can write the Laplace Transform of g(t) as:

L{g(t)} = g(t) e^{(-st) }dt

Substituting f(t-a) for g(t) and performing a change of variables t’ = t-a, we get:

L{f(t-a)} = f(t-a) e^{(-st)} dt

= f(t’) e^{(-s(t’+a))} dt’

= e^{(-as)} f(t’) e^{(-st’)} dt’

= e^{(-as) }F(s)

Therefore, the Second Shifting Property holds for any function f(t) with a Laplace Transform F(s).

Example 1: Find the Laplace Transform of the function f(t) = e^{(2t)}u(t-3).

Solution:

We can use the Second Shifting Property to find the Laplace Transform of f(t) by shifting the function by 3 units of time. Let g(t) = e^{(2t)}u(t-3). Then f(t) = g(t+3), and we have:

L{f(t)} = L{g(t+3)} = e^{(-3s)} L{g(t)}

We can find the Laplace Transform of g(t) using integration by parts:

L{g(t)} = e^{(2t)} e^{(-st)} dt

= e^{((2-s)t)} dt

= (1/(s-2)) e^{(2s)}

Substituting this result into the equation for L{f(t)}, we get:

L{f(t)} = e^{(-3s)} L{g(t)} = (e^{(-3s)}/(s-2)) e^{(2s)} = (e^{(2s-3s))}/(s-2)

**Describe Dirac-Delta(Impulse) Function**

The Dirac-Delta function, also known as impulse function or delta function, is a mathematical function used in signal processing, calculus, and other fields. It is defined as a distribution that is zero everywhere except at the origin, where it has an infinite value but a finite area of one. In this set of learning outcomes, we will describe the properties of the Dirac-Delta function and its applications.

Describe the Definition and Properties of the Dirac-Delta Function:

- The Dirac-Delta function is defined as a generalised function that has the following properties:

- It is zero for all values of x except at x = 0.
- The integral of the Dirac-Delta function over any finite interval is equal to one, i.e.,

integral of δ(x) dx from -∞ to ∞ = 1 - The Dirac-Delta function is an even function, i.e., δ(x) = δ(-x).
- The Dirac-Delta function has the sifting property, which states that it acts as an idealized point source. When it is multiplied by a test function f(x), the result is the value of f(x) at x=0, i.e.,

integral of f(x)δ(x) dx = f(0)

Derive the Laplace Transform of the Dirac-Delta Function:

- The Laplace transform of the Dirac-Delta function is obtained by integrating the function multiplied by e
^{(-st) }from -∞ to ∞, i.e.,

L{δ(t)} = [δ(t)e^{(-st)}] dt

Using the sifting property of the Dirac-Delta function, we get

L{δ(t)} = e^{(-s0)} = 1

Explain the use of Dirac-Delta Function in Solving Differential Equations:

- The Dirac-Delta function is used in solving differential equations in various fields of science and engineering. For example, it can be used to represent an instantaneous change in a system or a signal. Consider a simple first-order differential equation of the form:

dy/dt + ky = f(t)

where k is a constant and f(t) is a known function. The solution to this equation is given by

y(t) = y(0)e^{(-kt)} + [f(t)e^{(-kt})] dt from 0 to t

If f(t) is a Dirac-Delta function, then the integral reduces to f(0), and the solution becomes

y(t) = y(0)e^{(-kt) }+ f(0)e^{(-kt)}

This shows that the Dirac-Delta function can be used to model an instantaneous change in the system or a signal.

Discuss the use of Dirac-Delta Function in Signal Processing:

- The Dirac-Delta function is commonly used in signal processing, where it represents an impulse or a brief burst of energy. For example, a sudden loud noise can be modelled as a Dirac-Delta function. The convolution of a signal with a Dirac-Delta function results in a scaled and shifted version of the signal, which can be used to extract features or analyze the signal.

In summary, the Dirac-Delta function is a mathematical tool that is used to represent an idealised point source, and it has several important properties, including the sifting property and the Laplace transform. It has many applications in science and engineering, including the solution of differential equations and the analysis of signals in signal processing.

**Evaluate Laplace Transform of a Dirac-Delta (Impulse) Function**

The Laplace transform is a mathematical tool that is used to transform a function of time into a function of complex frequency. It is used in engineering, physics, and other fields to solve differential equations and analyze systems. In this set of learning outcomes, we will focus on the Laplace transform of the Dirac-Delta (Impulse) function.

Define the Laplace Transform:

- The Laplace transform of a function f(t) is defined as

F(s) = L{f(t)} = [f(t)e^{(-st)}] dt

where s is a complex frequency parameter.

Explain the Laplace Transform of the Dirac-Delta Function:

- The Laplace transform of the Dirac-Delta function δ(t) is obtained by integrating the function multiplied by e
^{(-st)}from -∞ to ∞, i.e.,

L{δ(t)} = [δ(t)e^{(-st)}] dt

Using the sifting property of the Dirac-Delta function, we get

L{δ(t)} = e^{(-s0)} = 1

Thus, the Laplace transform of the Dirac-Delta function is equal to 1.

Derive the Laplace Transform of the Derivative of the Dirac-Delta Function:

- The Laplace transform of the derivative of the Dirac-Delta function δ'(t) is obtained by integrating the function multiplied by e
^{(-st)}from -∞ to ∞, i.e.,

L{δ'(t)} = [δ'(t)e^{(-st)}] dt

Using integration by parts, we can show that

L{δ'(t)} = -sL{δ(t)}

Substituting L{δ(t)} = 1, we get

L{δ'(t)} = -s

Thus, the Laplace transform of the derivative of the Dirac-Delta function is equal to -s.

Evaluate the Laplace Transform of a Product of a Function and Dirac-Delta Function:

- The Laplace transform of a product of a function f(t) and the Dirac-Delta function δ(t) is obtained by integrating the product multiplied by e
^{(-st)}from 0 to ∞, i.e.,

L{f(t)δ(t)} = [f(t)δ(t)e^{(-st)}] dt

Using the sifting property of the Dirac-Delta function, we get

L{f(t)δ(t)} = f(0)

Thus, the Laplace transform of a product of a function and the Dirac-Delta function is equal to the value of the function at t=0.

Evaluate the Laplace Transform of the Product of a Function and the Derivative of Dirac-Delta Function:

- The Laplace transform of the product of a function f(t) and the derivative of the Dirac-Delta function δ'(t) is obtained by integrating the product multiplied by e
^{(-st)}from 0 to ∞, i.e.,

L{f(t)δ'(t)} = [f(t)δ'(t)e^{(-st)}] dt

Using integration by parts, we can show that

L{f(t)δ'(t)} = -f'(0)

**Describe Periodic Function**

A periodic function is a function that repeats its values at regular intervals or periods. The period is the smallest interval for which the function repeats itself. Periodic functions are encountered in many areas of mathematics, physics, and engineering. Some examples of periodic functions include the trigonometric functions such as sine and cosine, square wave function, sawtooth wave function, etc.

Definition of a Periodic Function:

- A function f(t) is said to be periodic with period T if, for all t in the domain of f(t), it satisfies the following condition:

f(t) = f(t + nT)

where n is an integer. This means that the value of the function at any point is the same as the value of the function at that point plus an integer multiple of the period T.

Characteristics of a Periodic Function:

- Some important characteristics of a periodic function are as follows:

- A periodic function has a constant period T, which is the smallest positive value for which the function repeats itself.
- A periodic function is symmetric about its period. This means that if we shift the function by one period, we get the same function.
- A periodic function can be decomposed into a series of sine and cosine functions using Fourier series.
- A periodic function may or may not have an average value over a period.
- A periodic function may have discontinuities or be non-differentiable at certain points.

- Examples of Periodic Functions:

- The sine and cosine functions are examples of periodic functions with a period of 2π.
- The sawtooth wave function is a periodic function with a period T that increases linearly from 0 to 2π and then jumps back to 0.
- The square wave function is a periodic function with a period T that takes on two values, +A and -A, alternately.
- The triangle wave function is a periodic function with a period T that increases linearly from 0 to A, then decreases linearly from A to -A, and then increases linearly from -A to 0.
- The periodic impulse train function is a periodic function that consists of a series of impulses spaced at regular intervals.

Applications of Periodic Functions:

- Periodic functions are used in many fields of science and engineering, including signal processing, acoustics, optics, and control systems. For example:

- In signal processing, periodic functions are used to represent signals such as audio and video signals, which have a repeating pattern.
- In acoustics, periodic functions are used to model the sound waves produced by musical instruments, which have a repeating pattern.
- In optics, periodic functions are used to model the properties of materials such as diffraction gratings, which have a repeating pattern.
- In control systems, periodic functions are used to represent the periodic disturbances that affect a system, such as vibrations in a mechanical system or power supply fluctuations in an electrical system.

**Evaluate Laplace Transform of Periodic Function**

The Laplace transform is a mathematical tool that is used to transform a function from the time domain to the frequency domain. It is a powerful technique that is used in many fields of engineering, including control systems, signal processing, and communications. In this learning outcome, we will learn how to evaluate the Laplace transform of a periodic function.

Definition of a Periodic Function:

- A periodic function is a function that repeats its values at regular intervals or periods. The period is the smallest interval for which the function repeats itself. Periodic functions are encountered in many areas of mathematics, physics, and engineering.

Laplace Transform of a Periodic Function:

- To find the Laplace transform of a periodic function, we use the following formula:

F(s) = (1 – e^{(-sT))}/s * f(t)

where F(s) is the Laplace transform of f(t), T is the period of the function, and s is the Laplace variable.

Example of Laplace Transform of a Periodic Function:

- Let’s consider the following periodic function:

f(t) = sin(2πt)

The period of this function is T = 1/2π.

To find the Laplace transform of this function, we use the formula:

F(s) = (1 – e^{(-sT))}/s * f(t)

F(s) = (1 – e^{(-s(1/2π)))}/s * sin(2πt)

Simplifying this equation, we get:

F(s) = (1 – e^{(-s/2π))}/s * sin(2πt)

This is the Laplace transform of the periodic function sin(2πt).

Properties of Laplace Transform of Periodic Functions:

- Some important properties of the Laplace transform of periodic functions are as follows:

- The Laplace transform of a periodic function is also a periodic function.
- The poles of the Laplace transform of a periodic function are located at s = -2πn/T, where n is an integer.
- The Laplace transform of a periodic function has an infinite number of poles on the negative real axis.
- The Laplace transform of a periodic function has an infinite number of zeros on the imaginary axis.

Applications of Laplace Transform of Periodic Functions:

- The Laplace transform of periodic functions is used in many fields of engineering, including control systems and signal processing. For example:

- In control systems, the Laplace transform of periodic functions is used to analyze the stability of a system under periodic disturbances.
- In signal processing, the Laplace transform of periodic functions is used to analyze the frequency content of a periodic signal.

Overall, understanding how to evaluate the Laplace transform of a periodic function is an important tool for engineers and scientists working in a wide range of fields.

**Describe Error Function**

The error function, denoted by erf(x), is a mathematical function that appears in probability theory, statistics, and various branches of applied mathematics. It is defined as the integral of a Gaussian function, which is a bell-shaped curve that represents a probability distribution. In this learning outcome, we will describe the error function and its properties.

Definition of Error Function:

- The error function is defined as follows:

erf(x) = (2/√π) e^{(-t2) }dt

where x is a real number.

The error function is an odd function, which means that erf(-x) = -erf(x).

Graphical Representation of Error Function:

- The error function is a smooth, continuous function that approaches -1 as x approaches negative infinity and approaches 1 as x approaches positive infinity. The graph of the error function is shown below:

[image source: https://en.wikipedia.org/wiki/Error_function]

Properties of Error Function:

- Some important properties of the error function are as follows:

- The error function is a non-elementary function, which means that it cannot be expressed in terms of elementary functions like polynomials, exponentials, and trigonometric functions.
- The error function is used to calculate the cumulative distribution function of a normal distribution. If X is a normally distributed random variable with mean μ and variance σ
^{2}, then the probability that X takes a value less than or equal to x is given by:

Pr(X ≤ x) = (1/2) [1 + erf((x-μ)/√(2σ^{2}))]

- The error function has a number of important special cases. For example, when x is small, we have:

erf(x) ≈ x

This is known as the small argument approximation.

Applications of Error Function:

- The error function has many applications in probability theory, statistics, and engineering. For example:

- In statistics, the error function is used to calculate the probability of a measurement falling within a certain range.
- In physics, the error function is used to describe the diffusion of particles in a medium.
- In engineering, the error function is used to model the probability of error in a communication system.

Overall, the error function is an important mathematical function that has a wide range of applications in various fields.

**Evaluate Laplace Transform of Error Function**

In this learning outcome, we will evaluate the Laplace transform of the error function, which is an important mathematical function that appears in probability theory, statistics, and engineering.

Definition of Laplace Transform:

- The Laplace transform is a mathematical operation that transforms a time-domain function f(t) into a complex frequency-domain function F(s). It is defined as follows:

F(s) = L[f(t)] = e^{(-st)} f(t) dt

where s is a complex number.

The Laplace transform has many applications in engineering and physics, such as solving differential equations and analyzing control systems.

Laplace Transform of Error Function:

- To evaluate the Laplace transform of the error function, we can use its definition and some algebraic manipulation. First, we write the definition of the error function as an integral:

erf(x) = (2/√π) e^{(-t2) }dt

Then, we multiply both sides by e^{(-sx)} and integrate from 0 to infinity:

e^{(-sx)} erf(x) dx = (2/√π) e(^{-sx)} e^{(-t2)} dt dx

We can swap the order of integration by applying Fubini’s theorem:

e^{(-sx)} erf(x) dx = (2/√π) e^{(-sx)} e^{(-t2)} dx dt

Now, we can simplify the inner integral by making the substitution u = x – t:

e^{(-sx)} erf(x) dx = (2/√π) e^{(-su)} e^{(-(x-u)2}) du dx

We can then evaluate the inner integral using the Gaussian integral:

e^{(-su)} e^{(-(x-u)2)} du = (π/2) e^{(s2/4 – sx)}

Substituting this back into the equation and simplifying, we get:

e^{(-sx)} erf(x) dx = (1/√s) e^{(s2/4)}

Therefore, the Laplace transform of the error function is:

L[erf(x)] = (1/√s) e^{(2/4)}

Applications of Laplace Transform of Error Function:

- The Laplace transform of the error function has many applications in engineering and physics. For example:

- In control systems, the Laplace transform of the error function is used to analyze the stability and performance of feedback control systems.
- In signal processing, the Laplace transform of the error function is used to analyze the frequency content of signals.
- In quantum mechanics, the Laplace transform of the error function is used to calculate the probability of a particle being in a certain state.

Overall, the Laplace transform of the error function is an important mathematical tool that has a wide range of applications in various fields.

**Describe Inverse Laplace Transform**

In this learning outcome, we will describe the inverse Laplace transform, which is an important mathematical operation that allows us to recover a time-domain function from its Laplace transform.

- Definition of Laplace Transform:

The Laplace transform is a mathematical operation that transforms a time-domain function f(t) into a complex frequency-domain function F(s). It is defined as follows:

F(s) = L[f(t)] = e^{(-st)} f(t) dt

where s is a complex number.

- Definition of Inverse Laplace Transform:

The inverse Laplace transform is the mathematical operation that allows us to recover a time-domain function f(t) from its Laplace transform F(s). It is defined as follows:

f(t) = L^{(-1})[F(s)] = (1/2πj) ∫γ-i∞^{γ}+i∞ e^{(st)} F(s) ds

where γ is a real number and the integration is taken along a vertical line in the complex plane.

The inverse Laplace transform can be thought of as the Laplace transform in reverse, where we go from the frequency domain back to the time domain.

Methods for Computing Inverse Laplace Transform:

- There are several methods for computing the inverse Laplace transform. Some common methods include:

- Partial Fraction Decomposition: This method involves decomposing a rational function into a sum of simpler fractions and then applying the inverse Laplace transform to each fraction. This method works well for functions with simple poles.
- Bromwich Integral: This method involves evaluating the inverse Laplace transform using a contour integral in the complex plane. This method works well for functions with poles and branch points in the left half of the complex plane.
- Convolution Integral: This method involves convolving the Laplace transform of a function with a known inverse Laplace transform, such as the Heaviside step function or the Dirac delta function. This method works well for functions that can be expressed as the product of two Laplace transforms.

Examples of Inverse Laplace Transform:

- Here are some examples of computing the inverse Laplace transform using different methods:

Example 1: Compute the inverse Laplace transform of F(s) = 1/(s+1)

Using partial fraction decomposition, we can write F(s) as:

F(s) = 1/(s+1) = A/(s+1)

where A = 1. Taking the inverse Laplace transform, we get:

f(t) = L^{(-1})[F(s)] = L^{(-1)}[A/(s+1)] = A e^{(-t)}

Therefore, the inverse Laplace transform of F(s) is f(t) = e^{(-t)}.

Example 2: Compute the inverse Laplace transform of F(s) = 1/(s^{2+1})

Using partial fraction decomposition, we can write F(s) as:

F(s) = 1/(s^{2+1}) = A/(s+i) + B/(s-i)

where A = B = 1/2. Taking the inverse Laplace transform, we get:

f(t) = L^{(-1)}[F(s)] = L^{(-1)}[A/(s+i) + B/(s-i)]

**Evaluate the Inverse Laplace Transform of Elementary Functions**

In this learning outcome, we will learn how to evaluate the inverse Laplace transform of some elementary functions using different methods.

- Inverse Laplace Transform of Elementary Functions:

- Laplace Transform of a Constant:

L{a} = a/s

- Inverse Laplace Transform of a Constant:

L^{(-1)}{a/s} = a

- Laplace Transform of Exponential Function:

L{e^{(at)}} = 1/(s-a)

- Inverse Laplace Transform of Exponential Function:

L^{(-1)}{1/(s-a)} = e^{(at)}

- Laplace Transform of Sinusoidal Function:

L{sin(at)} = a/(s^{2} + a^{2})

- Inverse Laplace Transform of Sinusoidal Function:

L^{(-1)}{a/(s^{2} + a^{2})} = sin(at)

- Laplace Transform of Sinusoidal Function:

L{cos(at)} = s/(s^{2 }+ a^{2})

- Inverse Laplace Transform of Sinusoidal Function:

L^{(-1}){s/(s^{2} + a^{2})} = cos(at)

- Laplace Transform of Step Function:

L{u(t)} = 1/s

- Inverse Laplace Transform of Step Function:

L^{(-1)}{1/s} = u(t)

- Laplace Transform of Ramp Function:

L{t} = 1/s^{2}

- Inverse Laplace Transform of Ramp Function:

L^{(-1)}{1/s^{2}} = t*u(t)

Methods for Evaluating Inverse Laplace Transform of Elementary Functions:

- There are different methods for evaluating the inverse Laplace transform of elementary functions. Some common methods include:

- Table Look-up: This method involves using a Laplace transform table to find the inverse Laplace transform of a function. This method works well for functions with simple Laplace transforms that are listed in the table.
- Partial Fraction Decomposition: This method involves decomposing a rational function into a sum of simpler fractions and then using the table look-up method to find the inverse Laplace transform of each fraction. This method works well for functions with simple poles.
- Bromwich Integral: This method involves evaluating the inverse Laplace transform using a contour integral in the complex plane. This method works well for functions with poles and branch points in the left half of the complex plane.

Examples of Evaluating Inverse Laplace Transform of Elementary Functions:

- Here are some examples of evaluating the inverse Laplace transform of elementary functions using different methods:

Example 1: Evaluate the inverse Laplace transform of F(s) = 1/(s+3)

To evaluate the inverse Laplace transform of F(s) = 1/(s+3), we can use the formula for the Laplace transform of e^{at}:

L{e^{at}} = 1/(s – a)

Therefore, we can write F(s) as:

F(s) = 1/(s+3) = 1/(s – (-3))

So, we can see that the inverse Laplace transform of F(s) is simply:

f(t) = L^{-1}{F(s)} = e^{(-3t)}

Therefore, the inverse Laplace transform of F(s) is f(t) = e^{(-3t)}.

Example 2: Evaluate the inverse Laplace transform of F(s) = (s+1)/(s^{2}+4s+5)

To evaluate the inverse Laplace transform of F(s) = (s+1)/(s^{2}+4s+5), we can use partial fraction decomposition to rewrite the function in terms of simpler terms that have known Laplace transforms.

First, we need to find the roots of the denominator of F(s), which are given by:

s^{2 }+ 4s + 5 = 0

Using the quadratic formula, we find that the roots are:

s = -2 + i and s = -2 – i

So, we can write the denominator of F(s) as:

s^{2} + 4s + 5 = (s – (-2 + i))(s – (-2 – i))

Next, we can write F(s) as:

F(s) = (s+1)/[(s – (-2 + i))(s – (-2 – i))]

Using partial fraction decomposition, we can write F(s) as:

F(s) = A/(s – (-2 + i)) + B/(s – (-2 – i))

where A and B are constants that we need to solve for.

Multiplying both sides of the equation by the denominator, we get:

(s+1) = A(s – (-2 – i)) + B(s – (-2 + i))

Now, we can solve for A and B by substituting in the roots of the denominator:

s = -2 + i:

(-2 + i + 1) = A((-2 + i) – (-2 – i))

-i = -2Ai

A = (1/2) + (1/2i)

s = -2 – i:

(-2 – i + 1) = B((-2 – i) – (-2 + i))

-i = 2Bi

B = -(1/2) + (1/2i)

Therefore, we can write F(s) as:

F(s) = [(1/2) + (1/2i)]/(s – (-2 + i)) + [-(1/2) + (1/2i)]/(s – (-2 – i))

Now, we can use the inverse Laplace transform to find f(t), which is given by:

f(t) = L^{-1}{F(s)}

Using the table of Laplace transforms, we find that the inverse Laplace transform of 1/(s – a) is e^{(at)}, so we have:

f(t) = [(1/2) + (1/2i)]e^{((-2+i)t)} + [-(1/2) + (1/2i)]e^{((-2-i)t)}

Simplifying the expression, we get:

f(t) = e^{(-2t)}(cos(t) + sin(t))

Therefore, the inverse Laplace transform of F(s) is f(t) = e^{(-2t)}(cos(t) + sin(t)).

Example 3: Evaluate the inverse Laplace transform of F(s) = (s+2)/(s^{2}+2s+10)

To evaluate the inverse Laplace transform of F(s) = (s+2)/(s^{2}+2s+10), we can use partial fraction decomposition to rewrite the function in terms of simpler terms that have known Laplace transforms.

First, we need to find the roots of the denominator of F(s), which are given by:

s^{2} + 2s + 10 = 0

Using the quadratic formula, we find that the roots are:

s = -1 + 3i and s = -1 – 3i

So, we can write the denominator of F(s) as:

s^{2} + 2s + 10 = (s – (-1 + 3i))(s – (-1 – 3i))

Next, we can write F(s) as:

F(s) = (s+2)/[(s – (-1 + 3i))(s – (-1 – 3i))]

Using partial fraction decomposition, we can write F(s) as:

F(s) = A/(s – (-1 + 3i)) + B/(s – (-1 – 3i))

where A and B are constants that we need to solve for.

Multiplying both sides of the equation by the denominator, we get:

(s+2) = A(s – (-1 – 3i)) + B(s – (-1 + 3i))

Now, we can solve for A and B by substituting in the roots of the denominator:

s = -1 + 3i:

(-1 + 3i + 2) = A((-1 + 3i) – (-1 – 3i))

3 + 3i = 6Ai

A = (1/2) – (1/2i)

s = -1 – 3i:

(-1 – 3i + 2) = B((-1 – 3i) – (-1 + 3i))

1 – 3i = -6Bi

B = -(1/2) + (1/2i)

Therefore, we can write F(s) as:

F(s) = [(1/2) – (1/2i)]/(s – (-1 + 3i)) + [-(1/2) + (1/2i)]/(s – (-1 – 3i))

Now, we can use the inverse Laplace transform to find f(t), which is given by:

f(t) = L^{-1}{F(s)}

Using the table of Laplace transforms, we find that the inverse Laplace transform of 1/(s – a) is e^{(at)}, so we have:

f(t) = [(1/2) – (1/2i)]e^{((-1+3i)t)} + [-(1/2) + (1/2i)]e^{((-1-3i)t)}

Simplifying the expression, we get:

f(t) = (1/2)(e^{(-t)}cos(3t) – e^{(-t)}sin(3t))

Therefore, the inverse Laplace transform of F(s) is f(t) = (1/2)(e^{(-t)}cos(3t) – e^{(-t)}sin(3t)).

**Evaluate the Inverse Laplace Transform of Functions Multiplied by s**

The Laplace transform is a mathematical technique used to transform a function from the time domain to the frequency domain. It is commonly used in engineering and science to solve differential equations and analyze systems. The inverse Laplace transform is the process of transforming a function from the frequency domain to the time domain.

When a function is multiplied by “s” before taking the Laplace transform, it is referred to as a “s-domain” function. The inverse Laplace transform of functions multiplied by “s” can be evaluated using the differentiation property of the Laplace transform. This property states that if the Laplace transform of a function f(t) is F(s), then the Laplace transform of tf(t) is -dF(s)/ds.

Examples:

- Evaluate the inverse Laplace transform of (s+3)/(s
^{2}+9).

Solution:

Let F(s) be the Laplace transform of f(t). Then,

F(s) = L{f(t)} = e^{(-st)} f(t) dt

Multiplying both sides by “s” gives:

sF(s) = L{tf(t)} = e^{(-st)} tf(t) dt

Taking the derivative of F(s) with respect to “s” gives:

dF(s)/ds = L{-tf(t)} = – e^{(-st)} tf(t) dt

Thus, the inverse Laplace transform of (s+3)/(s^{2}+9) is:

f(t) = -1/2(t cos 3t + 3 sin 3t)

- Evaluate the inverse Laplace transform of (2s+1)/(s
^{2}+4s+5).

Solution:

First, we factor the denominator:

s^{2}+4s+5 = (s+2)^{2}+1

Let F(s) be the Laplace transform of f(t). Then,

F(s) = L{f(t)} = e^{(-st)} f(t) dt

Multiplying both sides by “s” gives:

sF(s) = L{tf(t)} = e^{(-st)} tf(t) dt

Taking the derivative of F(s) with respect to “s” gives:

dF(s)/ds = L{-tf(t)} = – e^{(-st)} tf(t) dt

Thus, the inverse Laplace transform of (2s+1)/(s^{2}+4s+5) is:

f(t) = -1/2 e^{(-2t)} (t sin t + (2-t) cos t)

**Evaluate the Inverse Laplace Transform of functions Divided by s**

The Laplace transform is a mathematical tool used to transform a function from the time domain to the frequency domain. The inverse Laplace transform is the process of transforming a function from the frequency domain to the time domain. When a function is divided by “s” before taking the Laplace transform, it is referred to as an “s-domain” function.

The inverse Laplace transform of functions divided by “s” can be evaluated using the integration property of the Laplace transform. This property states that if the Laplace transform of a function f(t) is F(s), then the Laplace transform of f(τ) dτ is 1/s F(s).

Examples:

- Evaluate the inverse Laplace transform of 1/s.

Solution:

Let F(s) be the Laplace transform of f(t). Then,

F(s) = L{f(t)} = e^{(-st)} f(t) dt

Dividing both sides by “s” gives:

F(s)/s = L{ f(τ) dτ} = e^{(-st)} f(τ) dτ ds

Taking the inverse Laplace transform of both sides gives:

f(t) = L^{-1}{F(s)/s} = ∫[c-i∞, c+i∞] e^{(st)} F(s)/s ds

Using the residue theorem, we can show that the inverse Laplace transform of 1/s is:

f(t) = 1

- Evaluate the inverse Laplace transform of 1/(s
^{2}+9).

Solution:

Let F(s) be the Laplace transform of f(t). Then,

F(s) = L{f(t)} = e^{(-st)} f(t) dt

Dividing both sides by “s” gives:

F(s)/s = L{ f(τ) dτ} = e^{(-st)} f(τ) dτ ds

Taking the inverse Laplace transform of both sides gives:

f(t) = L^{-1}{F(s)/s} = ∫[c-i∞, c+i∞] e^{(st)} F(s)/s ds

Using partial fractions, we can write:

1/(s^{2}+9) = (1/3) (1/(s+3i) – 1/(s-3i))

Then, the inverse Laplace transform of 1/(s^{2}+9) is:

f(t) = (1/3) (e^{(3it)} – e^{(-3it)}) = (2/3) sin(3t)

**Evaluate the Inverse Laplace Transform of functions by using Partial Fraction Method**

Partial fraction method is a technique used to simplify a rational function into a sum of simpler fractions. This method is commonly used to evaluate the inverse Laplace transform of functions.

To use the partial fraction method to evaluate the inverse Laplace transform of a function, follow these steps:

- Factor the denominator of the rational function.
- Write the function as a sum of simpler fractions, where each fraction has a linear factor in the denominator. If a factor is repeated, include a term for each power of that factor, up to the multiplicity of the factor.
- Find the constants in each term by comparing the coefficients of like terms on both sides of the equation.
- Take the inverse Laplace transform of each term.
- Add the resulting functions together to get the final answer.

Examples:

- Evaluate the inverse Laplace transform of (s+1)/(s
^{2}+4s+3).

Solution:

Step 1: Factor the denominator

s^{2 }+ 4s + 3 = (s+1)(s+3)

Step 2: Write as a sum of simpler fractions

(s+1)/(s^{2}+4s+3) = A/(s+1) + B/(s+3)

Step 3: Find the constants

(s+1) = A(s+3) + B(s+1)

Let s=-3

-2 = -2A

A = 1

Let s=-1

0 = 2A + 2B

B = -1

Step 4: Take the inverse Laplace transform of each term

L^{-1}{A/(s+1)} = A e^{(-t)}

L^{-1}{B/(s+3)} = B e^{(-3t)}

Step 5: Add the resulting functions together

f(t) = A e^{(-t) }+ B e^{(-3t)}

f(t) = e^{(-t)} – e^{(-3t)}

- Evaluate the inverse Laplace transform of (s+2)/(s
^{2}+2s+2).

Solution:

Step 1: Factor the denominator

s^{2} + 2s + 2 = (s+1+i)(s+1-i)

Step 2: Write as a sum of simpler fractions

(s+2)/(s^{2}+2s+2) = A/(s+1+i) + B/(s+1-i)

Step 3: Find the constants

(s+2) = A(s+1-i) + B(s+1+i)

Let s=-1+i

1 = A(2i)

A = -i/2

Let s=-1-i

1 = B(-2i)

B = i/2

Step 4: Take the inverse Laplace transform of each term

L^{-1}{A/(s+1+i)} = Ae^{(-t)}sin(t)

L^{-1}{B/(s+1-i)} = Be^{(-t)}cos(t)

Step 5: Add the resulting functions together

f(t) = Ae^{(-t)}sin(t) + Be^{(-t)}cos(t)

f(t) = e^{(-t)} sin(t + π/4)

**Evaluate the Inverse Laplace Transform of functions by using Convolution Theorem**

The Inverse Laplace Transform is a mathematical tool used to convert a Laplace domain function into a time domain function. It is widely used in control theory, signal processing, and engineering to analyze and design complex systems. Convolution theorem is one of the powerful tools for finding inverse Laplace transform of the functions that are difficult to compute using other methods. In this learning outcome, we will learn how to evaluate the Inverse Laplace Transform of functions using the Convolution theorem.

Convolution theorem:

Convolution is an operation that combines two functions to produce a third function. The convolution theorem states that the inverse Laplace transform of the convolution of two functions is equal to the product of their individual Laplace transforms. Mathematically, it can be expressed as:

L^{-1}{F(s)G(s)} = f(t) * g(t)

where L^{-1} denotes the Inverse Laplace Transform operator, F(s) and G(s) are Laplace transforms of two functions f(t) and g(t) respectively, and * denotes the convolution operation.

Example 1:

Let’s consider the following Laplace transform:

F(s) = 1/(s^{2}+4s+5)

We want to find the inverse Laplace transform of F(s) using the convolution theorem. We can write F(s) as the product of two factors:

F(s) = 1/(s+2-1j)(s+2+1j)

Now, we can find the inverse Laplace transform of each factor using partial fraction decomposition:

L^{-1}{1/(s+2-1j)} = e^{(-2t)}sin(t)

L^{-1}{1/(s+2+1j)} = e^{(-2t)}cos(t)

Using the convolution theorem, we can find the inverse Laplace transform of F(s):

L^{-1}{F(s)} = L^{-1}{(1/(s+2-1j))(1/(s+2+1j))}

= e^{(-2t)}sin(t) * e^{(-2t})cos(t)

= 1/2 e^{(-4t)}sin(2t)

Hence, the inverse Laplace transform of F(s) is 1/2 e^{(-4t)}sin(2t).

Example 2:

Let’s consider the following Laplace transform:

F(s) = 1/(s^{2}+9)

We want to find the inverse Laplace transform of F(s) using the convolution theorem. We can write F(s) as the product of two factors:

F(s) = 1/[(s+3i)(s-3i)]

Now, we can find the inverse Laplace transform of each factor using partial fraction decomposition:

L^{-1}{1/(s+3i)} = e^{(-3t)}sin(3t)

L^{-1}{1/(s-3i)} = e^{(3t})sin(3t)

Using the convolution theorem, we can find the inverse Laplace transform of F(s):

L^{-1}{F(s)} = L^{-1}{(1/(s+3i))(1/(s-3i))}

= e^{(-3t)}sin(3t) * e^{(3t)}sin(3t)

= 1/2 sin^{2}(3t)

Hence, the inverse Laplace transform of F(s) is 1/2 sin^{2}(3t).

Conclusion:

The Convolution theorem is a powerful tool for evaluating the Inverse Laplace Transform of complex functions.

**Evaluate Laplace Transform of Derivatives and Integrals**

The Laplace transform is a mathematical tool used to convert a time-domain function into a frequency-domain function. The Laplace transform of a function f(t) is defined as F(s) = L[f(t)], where s is a complex variable. The Laplace transform has many applications in control theory, signal processing, and engineering. In this learning outcome, we will learn how to evaluate the Laplace transform of derivatives and integrals.

Laplace Transform of Derivatives:

Let f(t) be a continuous function with a continuous derivative f'(t) on the interval [0,∞). The Laplace transform of f'(t) is given by:

L[f'(t)] = sF(s) – f(0)

where F(s) is the Laplace transform of f(t), and f(0) is the initial value of f(t).

Example 1:

Let’s consider the following function:

f(t) = 2t + 1

We want to find the Laplace transform of f'(t). The derivative of f(t) is:

f'(t) = 2

Using the Laplace transform of derivatives formula, we can find the Laplace transform of f'(t):

L[f'(t)] = sL[f(t)] – f(0) = s(1/(s^{2})) – 1

= 1/s^{2} – 1

Hence, the Laplace transform of f'(t) is 1/s^{2} – 1.

Laplace Transform of Integrals:

Let f(t) be a continuous function with Laplace transform F(s) on the interval [0,∞). The Laplace transform of f(τ)dτ is given by:

L[ f(τ)dτ] = F(s)/s

Example 2:

Let’s consider the following function:

f(t) = 3t^{2} + 2

We want to find the Laplace transform of f(τ)dτ. The integral of f(t) is:

f(τ)dτ = t^{3} + 2t

Using the Laplace transform of integrals formula, we can find the Laplace transform of f(τ)dτ:

L[f(τ)dτ] = F(s)/s = (6/s^{4} + 2/s^{2})/s

= 6/s^{5} + 2/s^{3}

Hence, the Laplace transform of f(τ)dτ is 6/s^{5} + 2/s^{3}.

Conclusion:

The Laplace transform of derivatives and integrals are important tools in control theory, signal processing, and engineering. They are used to convert time-domain functions into frequency-domain functions, which make it easier to analyze and design complex systems. The formulas for the Laplace transform of derivatives and integrals are simple and can be easily applied to a wide range of functions.

**Evaluate Inverse Laplace Transform of Derivatives and Integrals**

The Laplace transform is a mathematical tool used to convert a time-domain function into a frequency-domain function. The inverse Laplace transform is the reverse process, which converts a frequency-domain function into a time-domain function. In this learning outcome, we will learn how to evaluate the inverse Laplace transform of derivatives and integrals.

Inverse Laplace Transform of Derivatives:

Let F(s) be a Laplace transform of a function f(t) with continuous derivatives of order up to n. The inverse Laplace transform of F(s) is given by:

L^{-1}[s^{n}F(s)] = (-1)^{n} d^{n}/dt^{n} f(t)

where d^{n}/dt^{n} f(t) denotes the nth derivative of f(t).

Example 1:

Let’s consider the following function:

F(s) = s/(s^{2} + 1)

We want to find the inverse Laplace transform of F(s). The derivative of F(s) is:

F'(s) = (s^{2 }– 1)/(s^{2} + 1)^{2}

Using the inverse Laplace transform of derivatives formula, we can find the inverse Laplace transform of F(s):

L^{-1}[sF(s)] = d/dt sin(t)

L^{-1}[s2F(s)] = -d^{2}/dt^{2} sin(t)

Applying the formula for the inverse Laplace transform of derivatives, we get:

L^{-1}[sF(s)] = -d/dt cos(t)

Hence, the inverse Laplace transform of F(s) is -d/dt cos(t).

Inverse Laplace Transform of Integrals:

Let F(s) be a Laplace transform of a function f(t) on the interval [0,∞). The inverse Laplace transform of F(s)/s is given by:

L^{-1}[F(s)/s] = f(τ)dτ

Example 2:

Let’s consider the following function:

F(s) = 2/(s^{2 }– 4)

We want to find the inverse Laplace transform of F(s)/s. The inverse Laplace transform of F(s) is:

L^{-1}[F(s)] = cosh(2t)/2

Using the inverse Laplace transform of integrals formula, we can find the inverse Laplace transform of F(s)/s:

L^{-1}[F(s)/s] = L^{-1}[F(s)]dτ

= cosh(2τ)/2 dτ

= sinh(2t)/4

Hence, the inverse Laplace transform of F(s)/s is sinh(2t)/4.

Conclusion:

The inverse Laplace transform of derivatives and integrals are important tools in control theory, signal processing, and engineering. They are used to convert frequency-domain functions into time-domain functions, which make it easier to analyze and design complex systems. The formulas for the inverse Laplace transform of derivatives and integrals are simple and can be easily applied to a wide range of functions.

**Find the solution of Linear Differential Equations by using Laplace Transform**

Linear differential equations are important in many areas of science and engineering, including control theory, circuit analysis, and physics. The Laplace transform is a powerful mathematical tool that can be used to solve linear differential equations. In this learning outcome, we will learn how to find the solution of linear differential equations using the Laplace transform.

Laplace Transform of a Differential Equation:

Consider a linear differential equation of the form:

an *y ^{(n)} + a_{(n-1)}y^{(n-1)} + … + a1y’ + a_{0}*y = f(t)

where y(t) is the unknown function, f(t) is the forcing function, and a_{0}, a_{1}, …, an are constants. We can take the Laplace transform of both sides of the equation and use the properties of the Laplace transform to obtain:

L[an *y ^{(n)}] + L[a_{(n-1)}y^{(n-1)}] + … + L[a1y’] + L[a_{0}*y] = L[f(t)]

where L[f(t)] is the Laplace transform of f(t), and L[y^{(k)}] is the Laplace transform of y^{(k)}, the kth derivative of y(t).

We can use the properties of the Laplace transform to simplify the left-hand side of the equation. For example, using the Laplace transform property L[y’] = sY(s) – y(0), where y(0) is the initial condition, we can rewrite the left-hand side of the equation as:

an *s ^{n}*Y(s) + a

_{(n-1)}

*s*Y(s) + (a

^{(n-1)}Y(s) + … + a1s_{0}– y(0))Y(s)

where Y(s) is the Laplace transform of y(t).

Solving for Y(s), we get:

Y(s) = [L[f(t)] + sum(a_{k}*s^{(n-k)}*Y(s))) / (an *s^{n} + a_{(n-1)}*s ^{(n-1)} + … + a1*s + a

_{0}– y(0))

This expression for Y(s) gives us the Laplace transform of the solution to the differential equation.

Inverse Laplace Transform of the Solution:

Once we have obtained the Laplace transform of the solution Y(s), we can use the inverse Laplace transform to obtain the solution y(t) in the time domain. This can be done using tables of Laplace transforms or by using the partial fraction decomposition method.

Example:

Let’s consider the following differential equation:

y” + 3y’ + 2y = e^{(-t)}

with initial conditions y(0) = 0 and y'(0) = 1.

Taking the Laplace transform of both sides of the equation, we get:

s^{2}*Y(s) – s*y(0) – y'(0) + 3s*Y(s) – 3y(0) + 2Y(s) = 1/(s+1)

Substituting the initial conditions, we get:

s^{2}*Y(s) + 3s*Y(s) + 2Y(s) = 1/(s+1) + s

Solving for Y(s), we get:

Y(s) = 1/(s+1) + s/(s+1) + 1/(s+2)

**Find the solution of Simultaneous Differential Equations by using Laplace Transform**

Simultaneous differential equations are a set of differential equations that involve more than one unknown function. The Laplace transform is a powerful mathematical tool that can be used to solve simultaneous differential equations. In this learning outcome, we will learn how to find the solution of simultaneous differential equations using the Laplace transform.

Laplace Transform of Simultaneous Differential Equations:

Consider a set of n linear differential equations of the form:

a_{ij}*y _{j}(k) + … + ai_{1}*y

_{1}

^{(k) }= f

_{i}(t), i = 1, 2, …, n

where y_{1}, y_{2}, …, yn are the unknown functions, f_{1}(t), f_{2}(t), …, fn (t) are the forcing functions, and a_{ij} are constants. We can take the Laplace transform of both sides of each equation and use the properties of the Laplace transform to obtain:

sum(a_{ij}*s ^{k}*Y

_{j}(s)) = F

_{i}(s), i = 1, 2, …, n

where Y_{j}(s) is the Laplace transform of y_{j}(t), F_{i}(s) is the Laplace transform of f_{i}(t), and k is the order of the highest derivative in the differential equation.

We can rewrite this system of equations in matrix form as:

A(s)*Y(s) = F(s)

where A(s) is an n x n matrix of coefficients, Y(s) is a column vector of the Laplace transforms of the unknown functions, and F(s) is a column vector of the Laplace transforms of the forcing functions.

Solving for Y(s), we get:

Y(s) = [A(s)]^{(-1)}*F(s)

where [A(s)]^{(-1)} is the inverse of the matrix A(s).

Inverse Laplace Transform of the Solution:

Once we have obtained the Laplace transform of the solution Y(s), we can use the inverse Laplace transform to obtain the solution y(t) in the time domain. This can be done using tables of Laplace transforms or by using the partial fraction decomposition method.

Example:

Let’s consider the following set of two differential equations:

y1” + 2y1′ – y2′ = 0

y2” – 2y1′ + 4y2 = e^{(-t)}

with initial conditions y1(0) = 1, y1′(0) = 0, y2(0) = 0, and y2′(0) = 1.

Taking the Laplace transform of both sides of each equation, we get:

s^{2}*Y1(s) – s*y1(0) – y1′(0) + 2s*Y1(s) – y2(0) – s*Y2(s) = 0

s^{2}*Y2(s) – s*y2(0) – y2′(0) – 2s*Y1(s) + 4Y2(s) = 1/(s+1)

Substituting the initial conditions, we get:

s^{2}*Y1(s) + 2s*Y1(s) – s*Y2(s) = s + 1

-2s*Y1(s) + s ^{2}*Y2(s) + 4Y2(s) = 1/(s+1)