# Thread: How to solve this non-homogenous second order ODE?

1. ## How to solve this non-homogenous second order ODE?

$y''-2y'+y=xe^xlnx$

I don't know what I should do because of the logarithm function, the exercise has come after introducing the undetermined coefficients methods, so I assume it should be solved that way but I don't know how.

2. ## Re: How to solve this non-homogenous second order ODE?

Originally Posted by Nikita2011
$y''-2y'+y=xe^xlnx$

I don't know what I should do because of the logarithm function, the exercise has come after introducing the undetermined coefficients methods, so I assume it should be solved that way but I don't know how.
The general solution has the form $y(x)=C_1(x)e^x+C_2(x)xe^x$ . Have you covered the method of variation of parameters?

3. ## Re: How to solve this non-homogenous second order ODE?

Originally Posted by FernandoRevilla
The general solution has the form $y(x)=C_1(x)e^x+C_2(x)xe^x$ . Have you covered the method of variation of parameters?
well, that's obviously the general solutions of the homogenous question, but how to find $y_p$.

I don't know what the method of variation of parameters is, but I solved it through the method that we take $y_p = v_1y_1 + v_2y_2$ and then we look for $v_1,v_2$. It was finally solved but I wonder why the author has put it in the section of undetermined coefficients method, is it possible to solve this with that method too? I doubt.

4. ## Re: How to solve this non-homogenous second order ODE?

It is possible to solve via the method of undetermined coefficients by trying a $y_p$ of the form

$y = ax^3 e^x \ln x + b x^3 e^x$.