Sample Problems

LAPLACE TRANSFORMS

 

1. Show from first principles (by using integration by parts) that the Laplace transform of is given by .

 

2. Find (from tables) the Laplace transforms of the following functions:

 

[Ans.].

 

3. Find (from tables) the inverse Laplace transforms of the following functions:

(i) ,

(ii) ,

(iii) ,

(iv) ,

(v) ,

(vi) .

[Ans. (i) , (ii) , (iii) , (iv) , (v) , (vi) ].

 

4. Find the Laplace transforms of the following expressions:

(i) ; ,

(ii) ; ,

(iii) ;

[Ans. (i) , (ii) , (iii) , where is the Laplace transform of ].

 

5. Solve the following differential equation using Laplace transforms

, .

[Ans. ].

 

6. Solve the following differential equation using Laplace transforms

, .

[Ans. ].

 

7. Solve the following differential equation using Laplace transforms

, .

[Ans. ].

 

8. Solve the following differential equation using Laplace transforms

, .

[Ans. ].

 

9. Solve the following differential equation using Laplace transforms

, ,

for some constant .

[Ans. ].