TITLE: 2nd order differential equation with constant variable?
QUESTION [0 upvotes]: I'm given the equation:
$$y'' + 9y = 18x^2 + 31$$
and I am asked to solve it. I know how to solve constant coefficients 2nd order DE, but I need a guide on the steps to solving this problem? I am stuck. Please help.

REPLY [0 votes]: Find your homogeneous solution by finding the eigenvalues, which are $\pm3i$
Therefore your homogeneous solution is  $$y_c= c_1 \cos x + c_2 \sin x$$
For the particular solution consider $$y_p= Ax^2+Bx+C$$ and plug into your equation to get $$y_p = 2x^2+3$$
Thus your solution is $$y= c_1 \cos x + c_2 \sin x +2x^2+3$$
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