Analytical solution of the ODE

Question

Consider the following boundary value problem. \begin{equation*} \begin{split} u=x^2u'+\frac{x^3}{3}u'', &\quad u(0)=A, \quad u(1)=B \end{split} \end{equation*} Is there any analytical method to solve the problem? According to WolframAlpha one can write the equation as a Sturm-Liouville equation as: \begin{equation*} \begin{split} (x^3u')'&=3u \end{split} \end{equation*} or as an Emden-Fowler one as: \begin{equation*} \begin{split} xu''+3u'=\frac{3u}{x^2} \end{split} \end{equation*} but I can find no methods to solve the problem analytically. I would be very grateful if someone explains some method to solve the problem.

Answer 1

Hint

$$u=x^2u'+\frac{x^3}{3}u''$$ $$u=\frac v x \quad \implies \quad \left(\frac{1}{x}+\frac{1}{3}\right) v-\frac{1}{3} x \left(x v''+v'\right)=0$$ looks like a Bessel differential equation.

Answer 2

I’m not sure if it works for this specific case, but think of this like a Bessel Function solution, in that we want to “guess” the solution to be some polynomial. So we choose an attempt to let the function be the sum from a constant to some finite power $n$ of $x$ (or if you want to find a harder class of solutions, which this might be a part of: use an infinite series denoted as a power series) with coefficients $c_0$ through $c_n$, even further if it is harder you can work the sum backwards to negative powers of $x$. Then solve for what the coefficients need to be in order for this ODE to be solved. It may not necessarily be a nice pattern, so it could look like it isn’t converging, which means you can approximate the ODE numerically by cutting off the power series at a specified $n$ with the particular accuracy you desire. Sorry if this seems convoluted, but if you need clarification just send a comment and I’ll try to respond with anything that might help make this clearer.