Unit 2 — Laplace Transform

Laplace Transform is an important mathematical technique used to convert differential equations into algebraic equations, making many engineering problems easier to solve.

1. Laplace Transform

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

\mathcal{L}\{f(t)\} = \int_0^\infty e^{-st} f(t)\, dt

If $\mathcal{L}\{f(t)\} = F(s)$ , then $F(s)$ is called the Laplace Transform of $f(t)$ .

Common Laplace Transforms

Constant Function

\mathcal{L}\{1\} = \frac{1}{s}

Power Function

\mathcal{L}\{t^n\} = \frac{n!}{s^{n+1}}

where $n$ is a non-negative integer.

Exponential Function

\mathcal{L}\{e^{at}\} = \frac{1}{s - a}

Sine Function

\mathcal{L}\{\sin at\} = \frac{a}{s^2 + a^2}

Cosine Function

\mathcal{L}\{\cos at\} = \frac{s}{s^2 + a^2}

Hyperbolic Sine

\mathcal{L}\{\sinh at\} = \frac{a}{s^2 - a^2}

Hyperbolic Cosine

\mathcal{L}\{\cosh at\} = \frac{s}{s^2 - a^2}

2. Existence Theorem

The Laplace Transform generally exists when the function $f(t)$ :

Is piecewise continuous on every finite interval.
Is of exponential order as $t \to \infty$ .

A function is said to be of exponential order if there exist constants $M > 0$ , $a$ , and $T$ such that:

|f(t)| \leq Me^{at} \qquad \text{for } t > T

Under these conditions, the Laplace Transform

\mathcal{L}\{f(t)\} = \int_0^\infty e^{-st} f(t)\, dt

exists for $s > a$ .

3. Properties of Laplace Transform

Linearity Property

If $\mathcal{L}\{f(t)\} = F(s)$ and $\mathcal{L}\{g(t)\} = G(s)$ , then:

\mathcal{L}\{af(t) + bg(t)\} = aF(s) + bG(s)

where $a$ and $b$ are constants.

First Shifting Theorem

If $\mathcal{L}\{f(t)\} = F(s)$ , then:

\mathcal{L}\{e^{at} f(t)\} = F(s - a)

Second Shifting Theorem

If $\mathcal{L}\{f(t)\} = F(s)$ , then:

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

where $u(t - a)$ is the unit step function.

Change of Scale

If $\mathcal{L}\{f(t)\} = F(s)$ , then:

\mathcal{L}\{f(at)\} = \frac{1}{a} F\left(\frac{s}{a}\right)

for $a > 0$ .

Differentiation in the $s$ -Domain

If $\mathcal{L}\{f(t)\} = F(s)$ , then:

\mathcal{L}\{t f(t)\} = -\frac{dF(s)}{ds}

More generally:

\mathcal{L}\{t^n f(t)\} = (-1)^n \frac{d^n F(s)}{ds^n}

Integration in the $s$ -Domain

If $\mathcal{L}\{f(t)\} = F(s)$ , then:

\mathcal{L}\left\{\frac{f(t)}{t}\right\} = \int_s^\infty F(u)\, du

when the required conditions are satisfied.

4. Laplace Transform of Derivatives and Integrals

Laplace Transform is especially useful for transforming derivatives into algebraic expressions.

First Derivative

If $\mathcal{L}\{f(t)\} = F(s)$ , then:

\mathcal{L}\{f'(t)\} = sF(s) - f(0)

Second Derivative

\mathcal{L}\{f''(t)\} = s^2 F(s) - s f(0) - f'(0)

Third Derivative

\mathcal{L}\{f'''(t)\} = s^3 F(s) - s^2 f(0) - s f'(0) - f''(0)

$n$ -th Derivative

The general formula is:

\mathcal{L}\{f^{(n)}(t)\} = s^n F(s) - s^{n-1} f(0) - s^{n-2} f'(0) - \cdots - f^{(n-1)}(0)

Laplace Transform of an Integral

g(t) = \int_0^t f(\tau)\, d\tau

then:

\mathcal{L}\left\{\int_0^t f(\tau)\, d\tau\right\} = \frac{F(s)}{s}

5. Unit Step Function

The unit step function is defined as:

u(t - a) = \begin{cases} 0, & t < a \\ 1, & t \geq a \end{cases}

It is useful for representing functions that begin or change at a particular time.

Laplace Transform of Unit Step Function

\mathcal{L}\{u(t - a)\} = \frac{e^{-as}}{s}

Second Shifting Theorem

For a function multiplied by a shifted unit step:

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

Representing a Piecewise Function

A piecewise function can often be represented using unit step functions.

For example, if

f(t) = \begin{cases} f_1(t), & 0 \leq t < a \\ f_2(t), & t \geq a \end{cases}

then it can be written as:

f(t) = f_1(t) + u(t - a)\bigl[f_2(t) - f_1(t)\bigr]

6. Laplace Transform of Periodic Functions

A function $f(t)$ is periodic with period $T$ if:

f(t + T) = f(t)

The Laplace Transform of a periodic function is:

\mathcal{L}\{f(t)\} = \frac{\displaystyle\int_0^T e^{-st} f(t)\, dt}{1 - e^{-sT}}

where $T$ is the period.

This formula is useful for finding the Laplace Transform of repeating or periodic signals and engineering systems.

7. Inverse Laplace Transform

The inverse Laplace Transform is used to recover $f(t)$ from $F(s)$ .

If $\mathcal{L}\{f(t)\} = F(s)$ , then:

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

Common methods include:

Partial fractions
Standard transform pairs
First shifting theorem
Second shifting theorem
Algebraic manipulation
Convolution theorem

Basic Inverse Transform Examples

If $F(s) = \dfrac{1}{s}$ , then:

\mathcal{L}^{-1}\left\{\frac{1}{s}\right\} = 1

If $F(s) = \dfrac{1}{s - a}$ , then:

\mathcal{L}^{-1}\left\{\frac{1}{s - a}\right\} = e^{at}

If $F(s) = \dfrac{a}{s^2 + a^2}$ , then:

\mathcal{L}^{-1}\left\{\frac{a}{s^2 + a^2}\right\} = \sin at

If $F(s) = \dfrac{s}{s^2 + a^2}$ , then:

\mathcal{L}^{-1}\left\{\frac{s}{s^2 + a^2}\right\} = \cos at

8. Convolution Theorem

The convolution of two functions $f(t)$ and $g(t)$ is defined as:

(f * g)(t) = \int_0^t f(\tau) g(t - \tau)\, d\tau

If $\mathcal{L}\{f(t)\} = F(s)$ and $\mathcal{L}\{g(t)\} = G(s)$ , then the convolution theorem states:

\mathcal{L}\{f * g\} = F(s) G(s)

Therefore:

\mathcal{L}^{-1}\{F(s) G(s)\} = \int_0^t f(\tau) g(t - \tau)\, d\tau

The convolution theorem is particularly useful when direct inverse Laplace transformation is difficult.

9. Applications of Laplace Transform

Laplace Transform can be used to solve ordinary differential equations and engineering problems.

Solving an Initial-Value Problem

Consider:

a y'' + b y' + c y = f(t)

with initial conditions $y(0) = y_0$ and $y'(0) = y_1$ .

Taking the Laplace Transform:

a \mathcal{L}\{y''\} + b \mathcal{L}\{y'\} + c \mathcal{L}\{y\} = \mathcal{L}\{f(t)\}

Using

\mathcal{L}\{y''\} = s^2 Y(s) - s y(0) - y'(0)

and

\mathcal{L}\{y'\} = s Y(s) - y(0)

we obtain:

a\bigl[s^2 Y(s) - s y_0 - y_1\bigr] + b\bigl[s Y(s) - y_0\bigr] + c Y(s) = F(s)

Collecting the terms containing $Y(s)$ :

Y(s) = \frac{F(s) + a s y_0 + a y_1 + b y_0}{a s^2 + b s + c}

The solution is then obtained using:

y(t) = \mathcal{L}^{-1}\{Y(s)\}

Simultaneous Differential Equations

Consider:

x'(t) = a x + b y + f(t)

and

y'(t) = c x + d y + g(t)

Taking the Laplace Transform converts the differential equations into algebraic equations involving:

X(s) = \mathcal{L}\{x(t)\}, \qquad Y(s) = \mathcal{L}\{y(t)\}

The resulting algebraic system can then be solved for $X(s)$ and $Y(s)$ .

Finally:

x(t) = \mathcal{L}^{-1}\{X(s)\}, \qquad y(t) = \mathcal{L}^{-1}\{Y(s)\}

Important Laplace Transform Formulas

Basic Transform Table

Function $f(t)$	Laplace Transform $F(s)$
$1$	$\dfrac{1}{s}$

Derivative Formulas

\mathcal{L}\{f'(t)\} = s F(s) - f(0)

\mathcal{L}\{f''(t)\} = s^2 F(s) - s f(0) - f'(0)

\mathcal{L}\{f^{(n)}(t)\} = s^n F(s) - \sum_{k=0}^{n-1} s^{n-1-k} f^{(k)}(0)

Integral Formula

\mathcal{L}\left\{\int_0^t f(\tau)\, d\tau\right\} = \frac{F(s)}{s}

First Shifting Theorem

\mathcal{L}\{e^{at} f(t)\} = F(s - a)

Second Shifting Theorem

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

Periodic Function

\mathcal{L}\{f(t)\} = \frac{\displaystyle\int_0^T e^{-st} f(t)\, dt}{1 - e^{-sT}}

Convolution

(f * g)(t) = \int_0^t f(\tau) g(t - \tau)\, d\tau

and

\mathcal{L}\{f * g\} = F(s) G(s)

What You'll Learn

By completing this unit, you will be able to:

Understand the definition of the Laplace Transform.
Determine when a Laplace Transform exists.
Apply important properties of Laplace Transform.
Calculate transforms of standard functions.
Find transforms of derivatives and integrals.
Work with unit step functions.
Calculate transforms of periodic functions.
Find inverse Laplace transforms.
Apply partial fraction techniques.
Use shifting theorems.
Apply the convolution theorem.
Solve ordinary differential equations using Laplace Transform.
Solve initial-value problems.
Solve simultaneous differential equations.
Apply Laplace Transform to engineering systems.

Key Takeaway

Laplace Transform provides a powerful method for converting differential equations into algebraic equations, simplifying the solution of initial-value problems, simultaneous equations, periodic functions, and many engineering systems.

Unit 2 — Laplace Transform

1. Laplace Transform

Common Laplace Transforms

Constant Function

Power Function

Exponential Function

Sine Function

Cosine Function

Hyperbolic Sine

Hyperbolic Cosine

2. Existence Theorem

3. Properties of Laplace Transform

Linearity Property

First Shifting Theorem

Second Shifting Theorem

Change of Scale

Differentiation in the sss-Domain

Integration in the sss-Domain

4. Laplace Transform of Derivatives and Integrals

First Derivative

Second Derivative

Third Derivative

nnn-th Derivative

Laplace Transform of an Integral

5. Unit Step Function

Laplace Transform of Unit Step Function

Second Shifting Theorem

Representing a Piecewise Function

6. Laplace Transform of Periodic Functions

7. Inverse Laplace Transform

Basic Inverse Transform Examples

8. Convolution Theorem

9. Applications of Laplace Transform

Solving an Initial-Value Problem

Simultaneous Differential Equations

Important Laplace Transform Formulas

Basic Transform Table

Derivative Formulas

Integral Formula

First Shifting Theorem

Second Shifting Theorem

Periodic Function

Convolution

What You'll Learn

Key Takeaway

Differentiation in the $s$ -Domain

Integration in the $s$ -Domain

$n$ -th Derivative