case study on Laplace transform By Panchal Kaushik M. Enrollment No:13040006035 Department: Civil Engineering Department

The Laplace Transforms TOPIC : Definition & Laplace transform of some elementary functions First shifting theorem Inverse Laplace transform Laplace transform of Integrals Multiplication by power of t Inverse Laplace transform derivation Division by t Convolution theorem Application to Differential equation

Topic -1 Definition : Let f(t) be a function of t defined by for all t 0 ; then the laplace transform of f(t) is denoted by ; and is defined as Provided that the integral exists ‘S’ is parameter which may be rael or complex. Note : (1) is clearly a function of s and is also written as i.e. (2)

Laplace transform of some elementary functions

2 First shifting theorem If then prove that Formula :

3 Inverse Laplace transform If then is called the inverse laplace transform of and is denoted by Formula :

E.x. Find inverse Laplace transform of

4 Laplace transform of integrals Theorem : if then E.x. Find the Laplace transform of solution

5 Multiplication by power of t Theorem : If then e.x. FIND

6 Inverse Laplace transform derivation Theorem : e.x. FIND

7 Division by t Theorem : if then e.x. FIND

8 Convolution theorem Theorem : if and then Apply convolution theorem Let and

By convolution theorem

9 Application to Differential equation Solution of ordinary linear differential equation: the Laplace transform can also be used to solve ordinary as well as partial differential equation. We shall apply this method to solve only ordinary linear differential equation with constant coefficients. The advantage of this method is that it gives the particular solution directly ; without the necessity of first finding the general solution and evaluating the arbitrary constants. Note

Method : take Laplace transform on both side of the given differential equation using initial conditions. This gives an algebraic equation. Solve the algebraic equation to get in term of S i.e. divide by the coefficient of ; getting as known function of S. Resolve this function of S in to partial fraction and take the inverse Laplace transform of both sides. This gives Y as a function of t which is the required solution satisfying the given conditions.

E.x.1: use transform method to solve ; given that . Here given equation is Taking the Laplace transform on both sides; Using initial conditions ; we get

now taking Inverse Laplace transform on both sides; Which is required solution.