find annhilator of right side and use undertermined coeffs t

find annhilator of right side and use undertermined coeff\'s to solve for particular

y\'\'-3y\'+2y=5cos2x+8sin2x

Solution

Let, D denote, d/dx

D(5 cos2x+8 sin 2x)=2(-5 sin(2x)+8 cos(2x))

D^2(5 cos(2x)+8 sin(2x))=4(-5cos(2x)-8 sin(2x))=-4 (5 cos(2x)+8 sin(2x))

(D^2+4)(5 cos(2x)+8 sin(2x))=0

So annihilator is :D^2+4

Corrsponding characteristic is:k^2+4=0

k=+-2i

SO general form of particular solution is

yp=A sin(2x)+B cos(2x)

yp\'\'=-4yp

Substituting gives

-4(A sin(2x)+B cos(2x))-6(A cos(2x)-B sin(2x))+2(A sin(2x)+B cos(2x))=5 cos(2x)+8 sin(2x)

(-2A+6B)sin(2x)+(-6A-2B)cos(2x) =5 cos(2x)+8 sin(2x)

Comparing coefficients gives

-2A+6B=8

-6A-2B=5

Solving for A and B gives

So,

yp=-23 sin(2x)/20+19 cos(2x)/20