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