How to integrate an ordinary differential equation around a singular point?

[wdlang]

i have a second order ordinary differential equation of f(x):

f''+(E-A(A+1)/x^2)f=0, where A is a positive integer, E is a real constant

the domain is [0, \infty).

the boundary condition is f(x=0)=0

since this is a linear equation, i only need to determine f up to a overall constant

how to do this numerically from the origin and outward?

we can prove that in the neighborhood of the origin, f is on the order of x^(A+1)

 

[wdlang]

a naive idea is to use the boundary condition mentioned above and the 4th-order Runge-Kutta method to do it

my concern is that in near the origin, the A(A+1)/x^2 term is too large

and therefore i have no idea whether the RK method will work, or if it does, how to control the error by shrinking the step length.

 

[kosovtsov]

If I do not mistaken, the solution

$$f(x) = C\sqrt{x}J_{A+1/2}(\sqrt{E}x)$$

where C is an arbitrary constant fit your ODE and the boundary condition f(x=0)=0.

 

[wdlang]

If I do not mistaken, the solution

$$f(x) = C\sqrt{x}J_{A+1/2}(\sqrt{E}x)$$

where C is an arbitrary constant fit your ODE and the boundary condition f(x=0)=0.

thanks a lot

however, i am not interested in the analytical solution

i want a general numerical method to deal with such singular points

the equation above is just an example, for which, luckily, an analytical solution is available