use method of undetermined coefficients to find general solu

use method of undetermined coefficients to find general solution y\'\'+y\'-2y=-5cos(t)

Solution

First we find general solution to associated homogeneous ode

y\'\'+y\'-2y=0

We look for :

y=exp(kt)
Substituting gives

k^2+k-2=0

k^2+2k-k-2=0

k(k+2)-(k+2)=0

k=1,k=-2

y=A exp(t)+B exp(-2t)

Now we look for particular solution

Based on inhomogenous part ie 5 cos(t) the guess for particular solution is

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

Subtituting gives

-A cos(t)-B sin(t)-A sin(t)+B cos(t)-2 A cos(t)-2B sin(t)=-5 cos(t)

(-3A+B)cos(t)+(-3B-A)sin(t)=-5 cos(t)

COmparing coefficient gives

A=-3B

-3A+B=-5

9B+B=-5

B=-1/2

A=3/2

y=A exp(t)+B exp(-2t)+3 cos(t)/2-sin(t)/2