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 
r2-r=0r (r-1) = 0r (r-1)= 0r = 0; r = 1.

Therefore, the complimentary solution is 
y = c1e0t + c2ety = c1 + c2et 

Step 2

Now, we determine the following determinants:
W = 1et0et=etW1=0ettan tet=-et tan t W2 =100tan t = tan t
We have g(x) = tan x. So, we calculate each of the ui. Therefore,
u'1 =-et tan tet= - tan tu1=-tan t dt = ln(|cos t|)u2' =  tan tet u2= tan tet dt  

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