Second order ODE with Dirichlet boundary condition

Question

Consider the second order ODE $$x^2\frac{d^2 y}{dx^2} + x\frac{dy}{dx} + (x^2 - \alpha^2)y = 0, x \in [-r, r],$$ with the Dirichlet boundary conditions $y(r) = 0, y(-r) = 0$, and $\alpha \in \mathbb{C}$. I am trying to find how $y$ depends on $r$. Of course it would be best if this ODE could be explicitly solved, but that seems hopeless to me. May be one can give some bounds on $y$ and derivatives of $y$ in terms of $r$? Where can I find a treatment of such problems?

Answer 1

$x^2\frac{d^2 y}{dx^2} + x\frac{dy}{dx} + (x^2 - \alpha^2)y = 0 \quad$ is a Bessel equation. The general solution is : $$y(x)=c_1 J_{\alpha}(x)+c_2 Y_{\alpha}(x)$$ where $J_{\alpha}(x)$ and $Y_{\alpha}(x)$ are the Bessel functions of order $\alpha$ and of the first and second kind respectively.

http://mathworld.wolfram.com/BesselFunction.html

The condition $y(0)=0$ implies $c_2=0$ because $Y_{\alpha}(0)$ is infinite. $$y(x)=c_1 J_{\alpha}(x)$$ http://mathworld.wolfram.com/BesselFunctionoftheFirstKind.html

The Bessel functions of first kind are even functions $ J_{\alpha}(-x)= J_{\alpha}(x)$ . So only one condition remains : $ J_{\alpha}(r)=0$

For example, given integer values of $\alpha$ the next table shows the corresponding values of $r$.

This table comes from : http://mathworld.wolfram.com/BesselFunctionZeros.html

To have a rough idea how $y(x)$ (in fact $J_{\alpha}(x)$ ) depends from $r$, interpolate in the table to find an approximate of $\alpha$ . If you want no other zero of the function in $0<x<r$ consider only the first line $k=1$. Then, with the approximate of $\alpha$ interpolate between the curves on the nest figure :

This figure comes from : http://mathworld.wolfram.com/BesselFunctionoftheFirstKind.html

Of course, for something less intuitive, numerical calculus is required.