2. case study on Laplace transformBy
Panchal Kaushik M.
Enrollment No:
Department: Civil Engineering Department

3. 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

4. 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)

5. Laplace transform of some elementary functions

6. 2 First shifting theorem
If then prove that
Formula :

7. 3 Inverse Laplace transform
If then is called the inverse laplace transform of and is denoted by
Formula :

8. E.x. Find inverse Laplace transform of

9. 4 Laplace transform of integrals
Theorem : if then
E.x. Find the Laplace transform of
solution

10. 5 Multiplication by power of t
Theorem : If then
e.x FIND

11. 6 Inverse Laplace transform derivation
Theorem :
e.x FIND

12. 7 Division by t
Theorem : if then
e.x FIND

13. 8 Convolution theorem Theorem : if and then Apply convolution theorem
Let and

14. By convolution theorem

15. 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

16. 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.

17. 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

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