1. Use the Laplace transform to work out the solution of the following initial value problem. - cost if 1/2

Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
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1. Use the Laplace transform to work out the solution of the following initial value problem.
- cost if T/2 <t< 3m/2
y" + 2y' + y = g(t) where y(0) = 1, y'(0) = 2, and g(t)
otherwise
Here are the steps:
(a) Sketch g(t).
(b) Express g(t) using unit step functions (Heaviside functions) and appropriately shifted trigonometric
functions, so that you can apply the Laplace transform.
(c) Find an expression for Y(s), the Laplace transform of y(t). It will have several terms, and each is
a product of a rational function and an exponential function.
(d) Apply the method of partial fractions to decompose the rational terms.
(e) Make sure each of the terms is now written, so that your know its Laplace inverse.
(f) Write down the solution for the inital problem.
Transcribed Image Text:1. Use the Laplace transform to work out the solution of the following initial value problem. - cost if T/2 <t< 3m/2 y" + 2y' + y = g(t) where y(0) = 1, y'(0) = 2, and g(t) otherwise Here are the steps: (a) Sketch g(t). (b) Express g(t) using unit step functions (Heaviside functions) and appropriately shifted trigonometric functions, so that you can apply the Laplace transform. (c) Find an expression for Y(s), the Laplace transform of y(t). It will have several terms, and each is a product of a rational function and an exponential function. (d) Apply the method of partial fractions to decompose the rational terms. (e) Make sure each of the terms is now written, so that your know its Laplace inverse. (f) Write down the solution for the inital problem.
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