1. The problem statement, all variables and given/known data I'm pretty sure this is a typo? http://gyazo.com/802746486cc68852e5384d5a12aed596 2. Relevant equations See the image ^. 3. The attempt at a solution I believe the theorem they're talking about, is that you can write the general solution of a second order ODE : [itex]L[y] = y'' + p(t)y' + q(t)y = 0[/itex] Where L is the differential operator on a solution y. In the form : c1y1 + c2y2 ⇔ y1, y2 were both linearly independent solutions to the homogeneous system. That is : [itex]W(y_1, y_2) ≠ 0[/itex] where W is the wronskian of y1, y2. So a bit of analysis here, when x > 0, y1 = y2 = x3, so that W(y1, y2) = 0. Also when x < 0, y1 = x3 and y2 = -y1. So W(y1, y2) = 0 as well in this case. Now, in part a) I showed that a linear combination of these two was indeed a solution by taking the required derivatives and blah blah, the boring stuff. Now it's mind boggling me that this can't be a typo because this is a counter-example to this theorem apparently, so I must be missing something crucial here. Any pointers on this?