Question

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.

Answer 1

Let
`y=C_1x+C_2x^3=C_1y_1+C_2y_2`. Then
`y_1` and
`y_2` are two fundamental, linearly independent solution that satisfy


`f(x,y_1,{y_1}',{y_1}'')=0`

`f(x,y_2,{y_2}',{y_2}'')=0`

Note that
`{y_1}'=1`, so that
`x{y_1}'-y_1=0`. Adding
`y''` doesn't change this, since
`{y_1}''=0`.

So if we suppose


`f(x,y,y',y'')=y''+xy'-y=0`

then substituting
`y=y_2` would give


`6x+x(3x^2)-x^3=6x+2x^3\\eq0`

To make sure everything cancels out, multiply the second degree term by
`-\frac{x^2}3`, so that


`f(x,y,y',y'')=-\frac{x^2}3y''+xy'-y`

Then if
`y=y_1+y_2`, we get


`-\frac{x^2}3(0+6x)+x(1+3x^2)-(x+x^3)=-2x^3+x+3x^3-x-x^3=0`

as desired. So one possible ODE would be


`-\frac{x^2}3y''+xy'-y=0\iff x^2y''-3xy'+3y=0`

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