Solving a Differential Equations Initial Value Problem

Question

Here's a link to the question. I'm trying to reduce the order of the differential equation by letting u = y' and u' = y'' but I've only gotten stuck later in the process when trying to find a fundamental matrix. Also, I should add that in my class we haven't learned Abel's Theorem or Variation of Parameters. Thank you for any help!

Answer 1

There's a typo in the question. The first equation should be set equal to $0$. You are given one solution to the homogeneous equation. Use reduction of order to find a second solution. Then the homogeneous (or complementary) solution will be $y_h = c_1e^{-x^2/2} + c_2f(x).$

Now you need a nonhomogeneous (or particular) solution, $y_p$. You don't need variation of parameters or Abel's theorem here since they gave you a strong hint. $y_p = x^2$ doesn't quite work, but I bet you can massage it a little. Then the final answer is $y=y_h+y_p.$