Internal problem ID [8890]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1311.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x \left (x^{2}+1\right ) y^{\prime \prime }+\left (2 x^{2}+1\right ) y^{\prime }-v \left (v +1\right ) x y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.039 (sec). Leaf size: 52
dsolve(x*(x^2+1)*diff(diff(y(x),x),x)+(2*x^2+1)*diff(y(x),x)-v*(v+1)*x*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} \hypergeom \left (\left [-\frac {v}{2}, \frac {1}{2}+\frac {v}{2}\right ], \left [\frac {1}{2}\right ], x^{2}+1\right )+c_{2} \sqrt {x^{2}+1}\, \hypergeom \left (\left [1+\frac {v}{2}, \frac {1}{2}-\frac {v}{2}\right ], \left [\frac {3}{2}\right ], x^{2}+1\right ) \]
✓ Solution by Mathematica
Time used: 0.066 (sec). Leaf size: 61
DSolve[-(v*(1 + v)*x*y[x]) + (1 + 2*x^2)*y'[x] + x*(1 + x^2)*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_2 G_{2,2}^{2,0}\left (-x^2| {c} \frac {1-v}{2},\frac {v+2}{2} \\ 0,0 \\ \\ \right )+c_1 \, _2F_1\left (-\frac {v}{2},\frac {v+1}{2};1;-x^2\right ) \\ \end{align*}