Find a particular solution to y25y20sin5tSolutionsin5t is a

Find a particular solution to y+25y=20sin(5t).

Solution

sin(5t) is a solution to the associated homogeneous ode ie y\'\'+25y=0

If this were not the case we would take the guess for particular solution to be

yp=A sin(5t)+ B cos(5t)

But since sin(5t) is solution of homogeneous ode so the guess becomes

yp=At sin(5t)+Bt cos(5t)

yp\'=A sin(5t)+B cos(5t)+5A t cos(5t)-5Bt sin(5t)

yp\'\'=10A cos(5t)-10 B sin(5t)-25 A t sin(5t)-25 Bt cos(5t)=10 A cos(5t)-10 B sin(5t)-25yp

yp\'\'+25yp=10 A cos(5t)-10 B sin(5t)=-20 sin(5t)

So, A=0, B=2

yp=2t cos(5t)