• 1. Using the method of undertermined coefficients, solve y" +y = g (t), whereif 0

Answered: • 1. Using the method of undertermined…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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• 1. Using the method of undertermined coefficients, solve y" +y = g (t), where
if 0<t<T,
TeT-, if t T,
satisfying y (0) = 0, y' (0) = 1. Assume that y and y are also continuous at t = T.
t,
g (t) =
%3D

Expert Solution

Step 1

For $0 \leq t \leq π$ ,

The Differential Equation becomes $y^{''} + y = t$

The characteristic equation for this differential equation and its roots are

$r^{2} + 1 = 0 \Rightarrow r^{2} = - 1 \Rightarrow r_{1} = i, r_{2} = - i$

The complementary solution is $y_{c} = c_{1} \cos t + c_{2} \sin t$

Here, $g (t) = t$ is a linear polynomial so the guess for particular solution is $y_{p} = A t + B$

Plugging into the differential equation we get,

$A t + B = t$

Equating the coefficients and solving

$A = 1, B = 0$

Thus we get, $y = y_{c} + y_{p} = c_{1} \cos t + c_{2} \sin t + t$

using $y (0) = 0, y^{'} (0) = 1$ , we get,

$y (t) = c_{1} \cos t + c_{2} \sin t + t \Rightarrow 0 = c_{1} + 0 \Rightarrow c_{1} = 0 y^{'} (t) = - c_{1} \sin t + c_{2} \cos t \Rightarrow 1 = c_{2}$

Thus $y = \sin t + t$ is the required solution of the differential equation.

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• 1. Using the method of undertermined coefficients, solve y" +y = g (t), where if 0