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Mathematical problems for self assessment

Sample Problems
LAPLACE TRANSFORMS

 

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

 

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) image placeholder,

(ii) image placeholder,

(iii) image placeholder,

(iv) image placeholder,

(v) image placeholder,

(vi) image placeholder.

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

 

4. Find the Laplace transforms of the following expressions:

(i)image placeholder ;image placeholder ,

(ii)image placeholder ;image placeholder ,

(iii)image placeholder ;image placeholder

[Ans. (i) image placeholder, (ii) image placeholder, (iii) image placeholder, where image placeholderis the Laplace transform of image placeholder].

 

5. Solve the following differential equation using Laplace transforms

image placeholderimage placeholder,image placeholder .

[Ans. image placeholder].

 

6. Solve the following differential equation using Laplace transforms

image placeholder,image placeholder .

[Ans. image placeholder].

 

7. Solve the following differential equation using Laplace transforms

image placeholder,image placeholder .

[Ans. image placeholder].

 

8. Solve the following differential equation using Laplace transforms

image placeholder,image placeholder .

[Ans. image placeholder].

 

9. Solve the following differential equation using Laplace transforms

image placeholder,image placeholder ,

for some constant image placeholder.

[Ans. image placeholder].