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

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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

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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}+B^{-t}

Now we must solve for the non-homogencous part:

y(t)=Ce^{t}, y'(t)=Ce^{t} ,  y''(t)=Ce^{t}

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

y''+2y'+y=2e^t

(Ce^{t})+2(Ce^{t})+(Ce^{t})=2e^t

4Ce^t=2e^t => C=1/2

So therefore the final solution is: 

y(t)=Ae^{-t}+B^{-t}+1/2
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