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Name: Apply variation of parameters to find a particular solution of y ''' -
Uploaded: 2024-03-20T06:46:42.000Z
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Description: Apply variation of parameters to find a particular solution of y ''' - y' = tan x

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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Apply variation of parameters to find a particular solution of y ''' - y' = tan x

Expert Solution

Step 1

The given differential equation is
$y'' - y' = \tan x$ .

We have to solve this using the variation of parameters method.

The auxiliary equation is
$r^{2} - r = 0 \Rightarrow r (r - 1) = 0 \Rightarrow r (r - 1) = 0 \Rightarrow r = 0; r = 1 .$

Therefore, the complimentary solution is
$y = c_{1} e^{0 t} + c_{2} e^{t} y = c_{1} + c_{2} e^{t}$

Step 2

Now, we determine the following determinants:
$W = |\begin{matrix} 1 & e^{t} \\ 0 & e^{t} \end{matrix}| = e^{t} W_{1} = |\begin{matrix} 0 & e^{t} \\ \tan t & e^{t} \end{matrix}| = - e^{t} \tan t W_{2} = |\begin{matrix} 1 & 0 \\ 0 & \tan t \end{matrix}| = \tan t$
We have $g (x) = \tan x$ . So, we calculate each of the $u_{i}$ . Therefore,
$u'_{1} = \frac{- e^{t} \tan t}{e^{t}} = - \tan t \Rightarrow u_{1} = - \int \tan t d t = \ln (| \cos t |) u_{2}^{'} = \frac{\tan t}{e^{t}} \Rightarrow u_{2} = \int \frac{\tan t}{e^{t}} d t$

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