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Laplace Transforms : Convolution Products and Differential Equations

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(1) Calculate the following convolution products :-

t t -t
(a) t * e (b) e * e

(c) sin t * sin t (d) H ( t - 1 ) * t²

Hint :- cos (A - B) - cos (A + B) = 2 sin A sin B.

(2) Find the solutions of the differential equations :-

(a) x + x = H ( t - 1 ) x (0) = 0

(b) x + x = 1 - H ( t - 1 ) x (0) = 0

(c) x + x = H ( t - 1 ) - H (t - 2 ) x (0) = 0

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Solution Summary

Convolution products are found and differential equations are solved. The solution is comprehensive.

Solution Preview

1) We know that the Laplace transform of a convolution product of two functions is the multiplication of the Laplace transform of them. Using this, we solve the first part:

(a) We will have:

which is the multiplication of the Laplace transforms of these functions. Now we must find the inverse of this Laplace transform. We must use partial fractions. That will give us:

and we can find that:
A=-1, B=-1, C=1.

Therefore, we will get:

and we can conclude that the result would be:

(b) Here we have:

which can be written as:

and we can find that:
A=1/2, ...

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