I am doing an exercise which asks for the concentration distribution of a given bacteria. After several computations I reached the following differential equation:
$$x^2y''-xy'+(1+x^2)y=0$$
which seem to be solved with Bessel but I have no idea how to do it. Could you help me please??
Letting $y(x)=x\,u(x)$,
$$y'=xu'+u,\\ y''=xu''+2u'.$$
Substituting into the ODE $x^2y'-xy'+(1+x^2)y=0$,
$$x^3u''+2x^2u'-x^2u'-xu+(1+x^2)xu=0,\\ \implies x^3u''+x^2u'+x^3u=0,\\ \implies x^2u''+xu'+x^2u=0.$$
This is the $0$-th order Bessel ODE, and so has general solution
$$u(x)=c_1J_0(x)+c_2Y_0(x).$$
Then the general solution to the original ODE is
$$y(x)=c_1xJ_0(x)+c_2xY_0(x).$$