Question

Use Method of Undetermined Coefficients te find a particular solution of the non-homogeneous equation. Find general solution of the non-homogeneous equation.

y''+2y'+5y=3sin(2t)

Answer 1

Find general solution of the non-homogeneous equation.

y''+2y'+5y=3sin(2t)

First solve the homogeneous equation:

y''+2y'+5y=0

=> r^2+2r+5=0 => r_{1}=-1+2i , r_{2}=-1-2i

So therefore, the general solution of the homogeneous part is:

y(t)=C_{1}e^{-t}cos2t+C_{2}e^{-t}sin2t

Now we must solve for the non-homogencous part:

y(t)=Asin2t+Bcos2t, y'(t)=2Acos2t-2Bsin2t, y''(t)=-4Asin2t-4Bcos2t

Plug the above into the rom-homogeneous equation and solve.

y''+2y'+5y=3sin(2t)

=>-4Asin2t-4Bcos2t+2Acos2t-2Bsin2t+Asin2t+Bcos2t=3sin2t

(A-4B)sin2t+(B+4A)cos2t=3sin2t

=> A-4B=3, B+4A=0

=> A=3/17  and  B= -12/17

So therefore the final solution is:

y(t)=C_{1}e^{-t}cos2t+C_{2}e^{-t}sin2t+3/17sin2t-12/17cos2t