Modified Helmholtz equation : perturbation theory I know that the equation : $$ x^2f''(x)+2xf'(x)+(x^2-n(n+1))f(x)=0 $$ is the Helmoholtz equation with solutions the spherical Bessel function. 
I wanted to know if there is a way to solve 
$$ x^2f''(x)+2xf'(x)+(x^2 (1+\alpha(x-x_0)/x_0)-n(n+1))f(x)=0 $$ with $(1+\alpha(x-x_0)/x_0)$ very small. I tried perturbation theory with $f(x)=B(x)+h(x)$ with $B$ the Bessel function and $h$ a perturbation for $n=0$ but even that I failed.
 A: What seems to be the problem? If you are prepared to use perturbation theory, then it's better to rewrite the equation as follows:
$$ f''(x)+\frac{2}{x} f'(x)+\left(a (x+b)-\frac{m}{x^2}\right)f(x)=0 $$ 
Where $a,b,m$ are the new parameters, and $a$ will be our small perturbation parameter.
Then expand:
$$f(x)=f_0(x)+a f_1(x)+a^2 f_2(x)+\cdots$$
We obtain:
$$f_0''(x)+\frac{2}{x} f_0'(x)-\frac{m}{x^2}f_0(x)=0$$
$$f_1''(x)+\frac{2}{x} f_1'(x)-\frac{m}{x^2}f_1(x)=-(x+b) f_0(x)$$
$$f_2''(x)+\frac{2}{x} f_2'(x)-\frac{m}{x^2}f_2(x)=-(x+b) f_1(x)$$
And so on. First equation gives us Bessel functions, all the other equations are also Bessel, but inhomogeneous. You will know the general homogeneous solution right away, so you will need to find some particular solution, since you will already know the source term.
It might get complicated analytically for large orders, but the set up is the same.

You can also try the Frobenius method to obtain the general power series solution for your equation, just as the Bessel functions are obtained.
