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'+y=2e^{-t}

Answer 1

Find general solution of the non-homogeneous equation.

y''+2y'+y=2e^{-t}

First solve the homogeneous equation:

y''+2y'+y=0

r^2+2r+1=0 <=> (r+1)(r+1)=0 

so r=-1 (repeated root)

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

y(t)+Ae^{-t}+Bt^{-t}

Now we must solve for the non-homogencous part:

y(t)=Ct^{2}e^{-t} , y'(t)=C(2te^{-t}-t^{2}e^{-t})

y''(t)=C(2e^{-t}-4te^{-t}+t^2e^{-t})

Plug the above into the rom-homogeneous equation and solve

y''+2y'+y=2e^{-t}

=> C(2e^{-t}-4te^{-t}+t^2e^{-t})+C(2te^{-t}-t^{2}e^{-t})+Ct^{2}e^{-t} =2e^{-t}

Then C=1

So therefore the final solution is:

y(t)+Ae^{-t}+Bt^{-t}+t^2e^{-t}