Second order inhomogeneous differentiel equation.

[Lindsayyyy]

Hi everyone,

Homework Statement

I shall find the solution for the following differential equation:
$$y''(x)+2y'(x)+y(x)=x^{2}+3$$

Homework Equations

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The Attempt at a Solution

At first solved the homogenous equation and found the general solution for the homogenous as the following:

$$y(x)=e^{-x}+xe^{-x}$$

Now I have to find an ansatz function I guess, but I don't know how to do this. I tried At³ but didn't work. Can someone help me?

Thanks in advance

 

[Quinzio]

Try
$$y=ax^2+bx+c$$

 

[Lindsayyyy]

Thanks, I guess that worked if I've done it correctly

 

[vela]

It's easy enough to check. Just plug your solution into the LHS of the equation and see if you get the RHS when you simplify.