Find a particular solution of the differential equation y2yy

Find a particular solution of the differential equation y\'\'-2y\'+y=cosx

Solution

Based on inhomogeneous part ie cos(x) the guess for particular solution is

yp= a sin(x)+b cos(x)

yp\'=a cos(x)-b sin(x)

yp\'\'=-yp

Substituting gives

-yp-2(a cos(x)-b sin(x))+yp=cos(x)

Hence comparing coefficients gives

b=0

a=-1/2

So ,

yp=-sin(x)/2