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Unit-2TransformsQuestion-1: Find the Laplace transform of the following functions-1.

2.

Sol. 1. Here

So that we can write it as-

Now-

2. Since

Or

Now-

Question-2: Find the Laplace transform of (1 + cos 2t)

Sol.

So that-

Question-3: Find the inverse Laplace transform of-

Sol.

Question-4: Find the inverse Laplace transform of-

Sol.

Question-5: Find the inverse transform of-

Sol.First we will convert it into partial fractions-

Qustion-6: Find the Laplace transform of t sin at.Sol. Here-

Question-7: Find the Laplace transform of

Sol. Here-

So that-

As we know that-

So that-

Hence-

Question-8: Find the Laplace transform of

.Sol. Here-

Now-

Question-9: Use Laplace transform method to solve the following equation-

Sol. Here we have-

Take Laplace transform of both sides, we get-

It becomes-

(

So that-

Now breaking it into partial fractions-

We get the following results on inversion-

Question-10: Using complex form find the Fourier series of the function f(x) = x2, defined on te interval [-1,1]Solution:Here the half-period is L=1.Therefore, the co-efficient c0 is,

For n

Integrating by parts twice,we obtain

=

=

= .

= .

Question-11: Find the Fourier transform of

Sol. As we know that the Fourier transform of f(x) will be-

So that-

Now put

So that-

Question-12: Find the Fourier sine transform of

Sol. Here x being positive in the interval (0, ∞)

Fourier sine transform of

will be-

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