Internal problem ID [8801]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1221.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{2} y^{\prime \prime }+\left (x -2 x^{2} f \relax (x )\right ) y^{\prime }+\left (x^{2} \left (1+f \relax (x )^{2}-f^{\prime }\relax (x )\right )-x f \relax (x )-v^{2}\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.004 (sec). Leaf size: 53
dsolve(x^2*diff(diff(y(x),x),x)+(x-2*x^2*f(x))*diff(y(x),x)+(x^2*(1+f(x)^2-diff(f(x),x))-x*f(x)-v^2)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} {\mathrm e}^{-\frac {\left (\int \frac {-2 f \relax (x ) x +1}{x}d x \right )}{2}} \sqrt {x}\, \BesselJ \left (v , x\right )+c_{2} {\mathrm e}^{-\frac {\left (\int \frac {-2 f \relax (x ) x +1}{x}d x \right )}{2}} \sqrt {x}\, \BesselY \left (v , x\right ) \]
✓ Solution by Mathematica
Time used: 0.017 (sec). Leaf size: 31
DSolve[y[x]*(-v^2 - x*f[x] + x^2*(1 + f[x]^2 - Derivative[1][f][x])) + (x - 2*x^2*f[x])*y'[x] + x^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to (c_1 J_v(x)+c_2 Y_v(x)) \exp \left (\int _1^xf(K[1])dK[1]\right ) \\ \end{align*}