Find a linear second-order differential equation f(x, y, y', y'') = 0 for which y = c1x + c2x3 is a two-parameter family of solutions. make sure that your equation is free of the arbitrary parameters c1 and c2.

Let . Then and are two fundamental, linearly independent solution that satisfy




Note that , so that . Adding doesn't change this, since .

So if we suppose



then substituting would give



To make sure everything cancels out, multiply the second degree term by , so that



Then if , we get



as desired. So one possible ODE would be



(See "Euler-Cauchy equation" for more info)