Laplace Transforms : Convolution Products and Differential Equations
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LAPLACE TRANSFORMS
(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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