# Mathematical problems for self assessment

###### 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. ].