Laplace Transform Study Notes

UNIT-V LAPLACE TRANSFORM

5.1 INTRODUCTION

• Definition of Transformation: A transformation is an operation that converts a mathematical expression to a different but equivalent form. For instance, logarithms simplify multiplication and division into addition and subtraction.

• Laplace Transform: A powerful mathematical technique for solving linear equations with given initial conditions through algebraic methods. Also applicable for:
    - Systems of differential equations
    - Partial differential equations
    - Integral equations

• Chapter Covered: Includes definition, properties of Laplace transform, and derivation of transforms for common functions in linear differential equations.

5.2 LAPLACE TRANSFORM

• Definition: For a function $f(t)$ defined for $t \geq 0$, the Laplace transform is given by:
    $L[f(t)] = rac{1}{s} imes ext{where } f(t)$ exists, denoted by:
    $L[f(t)] = F(s) = \int_0^{\infty} e^{-st} f(t) dt$
    - Parameters:
      - $s$: May be real or complex.
      - $F(s)$: Function of $s$.

• Piecewise Continuous Function: A function $f(t)$ is piecewise continuous on an interval $[a, b]$ if it can be divided into finitely many subintervals where the function is continuous and has finite left and right limits.

• Exponential Order: A function $f(t)$ is said to be of exponential order if:
    $\lim_{t \rightarrow \infty} e^{-st} f(t)$ is finite, where s > 0 exists.

Example 5.1

• Show that the function $f(t) = e^{t^3}$ is not of exponential order:
    $\lim_{t \rightarrow \infty} e^{-s t} e^{t^3} = \lim_{t \rightarrow \infty} e^{t^3 - st} = e^{\infty} = \infty$
    - Hence, $f(t) = e^{t^3}$ is not of exponential order.

Sufficient Conditions for Existence of Laplace Transform

The Laplace transform of $f(t)$ exists if:

1. $f(t)$ is piecewise continuous on $[a, b].$

2. $f(t)$ is of exponential order.

Note: These conditions are sufficient, not necessary.

Example 5.2

• Prove that the Laplace transform of $e^{t^2}$ does not exist:
    $\lim_{t \rightarrow \infty} e^{-s t} e^{t^2} = \lim_{t \rightarrow \infty} e^{-st + t^2} = e^{\infty} = \infty$
    - Therefore, Laplace transform of $e^{t^2}$ does not exist.

5.3 PROPERTIES OF LAPLACE TRANSFORM

Property 1: Linear Property

• For constants $a$ and $b$,
    $L[a f(t) \pm b g(t)] = a L[f(t)] \pm b L[g(t)]$
    - Proof:
      - By definition:
        $L[a f(t) \pm b g(t)] = \int_0^{\infty} e^{-st}[a f(t) \pm b g(t)] dt$
        $= a \int_0^{\infty} e^{-st}f(t) dt \pm b \int_0^{\infty} e^{-st}g(t) dt$
        $= a L[f(t)] \pm b L[g(t)]$

Property 2: Change of Scale Property

• If $L[f(t)] = F(s)$, then:
    L[f(a t)] = \frac{1}{a} F(sa), \, a > 0
    - Proof:
      - Starting from the definition:
        $L[f(at)] = \int_0^{\infty} e^{-st} f(at) dt$
        - Change variables, let $x = at$; thus $dt = \frac{dx}{a}$,
        $L[f(at)] = \int_0^{\infty} e^{-s\frac{x}{a}} f(x) \frac{dx}{a} = \frac{1}{a} \int_0^{\infty} e^{-\frac{s}{a} x} f(x) dx$
        - Therefore:
          $L[f(at)] = \frac{1}{a} F(s/a)$

Property 3: First Shifting Property

• If $L[f(t)] = F(s)$, then:
    1. $L[e^{-at} f(t)] = F(s + a)$
    2. $L[e^{at} f(t)] = F(s - a)$
    - Proof:
      - Proof for i): Starting from the definition,
        $L[e^{-at} f(t)] = \int_0^{\infty} e^{-st} e^{-at} f(t) dt = \int_0^{\infty} e^{-(s+a)t}f(t) dt = F(s + a)$
      - Proof for ii): Similar approach results in:
        $L[e^{at} f(t)] = \int_0^{\infty} e^{(a - s)t} f(t) dt = F(s - a)$

Property 4: Laplace Transforms of Derivatives

• For the first derivative,
    $L[f'(t)] = s L[f(t)] - f(0)$
    - Proof:
      - By integration by parts,
        $L[f'(t)] = \int_0^{\infty} e^{-st} f'(t) dt$
        - Taking the boundary terms and integrating shows:
          $= 0 - f(0) + s L[f(t)] = s L[f(t)] - f(0)$

Property 5: Laplace Transform of Derivative of Order n

• For the nth derivative:
    $L[f^{(n)}(t)] = s^n L[f(t)] - s^{n-1} f(0) - s^{n-2} f'(0) - \ldots - f^{(n-1)}(0)$
    - Proof: Induction leads through the previous property, consistently applying the recurrence relation to show.

5.4 LAPLACE TRANSFORM OF DERIVATIVES AND INTEGRALS

Problems Using the Formula

• Given:
    $L[t f(t)] = - \frac{d}{ds} L[f(t)]$

• Example 5.8: Find the Laplace transform for $t ext{sin}(4t)$
    - Solution applies derivative rules each step; result is derived systematically.

Exercise Solutions

• Calculate specific Laplace transforms and integrals as exercises, using the derived properties, such as the linear and derivative properties in transformations with known functions.