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Use the "guessing" method to solve the following 2nd-order inhomogeneous ODEs. Note - you may need to consider adding two guesses based on the inhomogeneous part. (a) d²y dr² dy +4 + 5y = sin(2x), dr

Use the "guessing" method to solve the following 2nd-order inhomogeneous ODEs. Note - you may need to consider adding two guesses based on the inhomogeneous part. (a) d²y dr² dy +4 + 5y = sin(2x), dr

Advanced Engineering Mathematics
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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Use the "guessing" method to solve the following 2nd-order inhomogeneous ODEs. Note - you may need to consider adding two guesses based on the inhomogeneous part. (a) (b) (c) d²y dy +4 +5y = sin (2x), dr² dx d²y dy +2=⁹ dr² dx d'y dz² dy dz = 12x², + 2y = e²+x.
Transcribed Image Text:Use the "guessing" method to solve the following 2nd-order inhomogeneous ODEs. Note - you may need to consider adding two guesses based on the inhomogeneous part. (a) (b) (c) d²y dy +4 +5y = sin (2x), dr² dx d²y dy +2=⁹ dr² dx d'y dz² dy dz = 12x², + 2y = e²+x.
Expert Solution
Step 1: Finding homogeneous solution

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Given nonhomogeneous differential equation

d2ydx2+4dydx+5y=sin⁡(2x)

Corresponding homogeneous equation is

d2ydx2+4dydx+5y=0

Let y=emx be trial solution then 

m2emx+4memx+5emx=0

So the auxiliary equation is

m2+4m+5=0

Solving, m=−2±i

Thus the complementary solution is

yh=e−2x(C1sin⁡(x)+C2cos⁡(x))

Where C1,C2 are constants.


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