3rd order homogeneous differential equation with non-constant coefficients

Question

How do I solve the following homogeneous differential equation?

$$x^3u'''-3x^2u''-x(x^2-6)u'+(x^2-6)u=0$$

I really have no idea where to start.

Answer 1

Hint: Let:

$$u(x) = x v(x)$$

• Calculate the first, second and third derivative of $u(x)$.
• Substitute back into ODE and simplify.
• This will reduce the ODE to a form you can work with.