Solve ď²y dx² 3 + y = secºx.

Algebra: Structure And Method, Book 1
(REV)00th Edition
ISBN:9780395977224
Author:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole
Publisher:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole
Chapter9: Systems Of Linear Equations
Section9.2: The Substitution Method
Problem 10OE
Question
Exercises (Reduction of order)
1. Solve
ď²y
dx²
3
+ y = sec x.
Transcribed Image Text:Exercises (Reduction of order) 1. Solve ď²y dx² 3 + y = sec x.
Expert Solution
Step 1: Solution

d2ydx2+y=sec3x

The C.F of the differential equation is c1sinx+c2cosx

Now finding the particular integral

The particular integral is 

P.I=u1v1+u2v2u1=sinxu2=cosxv1=-u2sec3xWdxv2=u1sec3xWdxW=sinxcosxcosx-sinx    =-1v1=cosxsec3xdx     =sec2x     =tanxv2=u1sec3xWdx     =-sinxcos3xdx     =-tanxsec2xdx

Let secx=y, dy=tanxsecxdx

    =-ydy=-y22=-sec2x2

P.I=tanxsinx-cosxsec2x2           =tanxsinx-secx2

 

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