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    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

    © BrainMass Inc. brainmass.com October 9, 2019, 3:50 pm ad1c9bdddf
    https://brainmass.com/math/partial-differential-equations/laplace-transforms-convolution-products-differential-equations-14410

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    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, ...

    Solution Summary

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

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