Find a particular solution for a non-homogeneous differential equation

[blouqu6]

Find a particular solution for the following equation:

y"+2y'+y=12.5e^-t

I'm not sure on which method to use. Here's my attempt using the undetermined coefficients method:

→y"+2y'+y=12.5e^-t
r²+2r+1=0
r=-1 *not even sure if this part is useful

→y_p=e^-t
y_p'=-e^-t
y_p"=e^-t

→y"+2y'+y=12.5e^-t
(e^-t)+2(-e^-t)+(e^-t)=12.5e^-t

e^-t(1-2+1)=e^-t(12.5)

This is where I get stuck; I don't believe I'm taking the right approach from the get-go. Any help would be greatly appreciated.

 

[Mark44]

Find a particular solution for the following equation:

y"+2y'+y=12.5e^-t

I'm not sure on which method to use. Here's my attempt using the undetermined coefficients method:

→y"+2y'+y=12.5e^-t
r²+2r+1=0
r=-1 *not even sure if this part is useful

It's very useful, as it gives you the complementary solution, the solution to the homogeneous problem. Note that r = -1 is a repeated solution of the characteristic equation.

→y_p=e^-t

Not a good choice for a particular solution. e^-t is a solution of the homogeneous problem, so can't possibly be a solution of the nonhomogeneous problem.

y_p'=-e^-t
y_p"=e^-t

→y"+2y'+y=12.5e^-t
(e^-t)+2(-e^-t)+(e^-t)=12.5e^-t

e^-t(1-2+1)=e^-t(12.5)

This is where I get stuck; I don't believe I'm taking the right approach from the get-go. Any help would be greatly appreciated.