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\'\'-6y=5xe^x

Solution

Let, D denote d/dx

D(5xe^x)=5e^x+5xe^x

So,

D(5xe^x)-5xe^x=5e^x

(D-1)(5xe^x)=5e^x

(D-1)^1(5xe^x)=(D-1)(5e^x)=0

So, annihilator is (D-1)^2

HEnce the form for particular solutino is

yp=e^x(Ax+B)

Substituting gives

(e^x(Ax+B))\'\'-e^x(Ax+B)=5xe^x

e^x(Ax+2A+B)-6e^x(Ax+B)=5xe^x

e^x(-5Ax+2A-5B)=5xe^x

So, A=-1

2A-5B=0

5B=2A=-2

B=-2/5

So, particular solution is

yp=e^x(-x-2/5)