Internal problem ID [8897]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1318.
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 (a \,x^{2}+b \right ) y^{\prime }+c x y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 122
dsolve(x*(x^2-1)*diff(diff(y(x),x),x)+(a*x^2+b)*diff(y(x),x)+c*x*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} \hypergeom \left (\left [-\frac {1}{4}+\frac {a}{4}+\frac {\sqrt {a^{2}-2 a -4 c +1}}{4}, -\frac {1}{4}+\frac {a}{4}-\frac {\sqrt {a^{2}-2 a -4 c +1}}{4}\right ], \left [-\frac {b}{2}+\frac {1}{2}\right ], x^{2}\right )+c_{2} x^{b +1} \hypergeom \left (\left [\frac {1}{4}+\frac {a}{4}+\frac {b}{2}+\frac {\sqrt {a^{2}-2 a -4 c +1}}{4}, \frac {1}{4}+\frac {a}{4}+\frac {b}{2}-\frac {\sqrt {a^{2}-2 a -4 c +1}}{4}\right ], \left [\frac {3}{2}+\frac {b}{2}\right ], x^{2}\right ) \]
✓ Solution by Mathematica
Time used: 0.139 (sec). Leaf size: 144
DSolve[c*x*y[x] + (b + a*x^2)*y'[x] + x*(-1 + x^2)*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 \, _2F_1\left (\frac {1}{4} \left (a-\sqrt {(a-1)^2-4 c}-1\right ),\frac {1}{4} \left (a+\sqrt {(a-1)^2-4 c}-1\right );\frac {1-b}{2};x^2\right )+i e^{\frac {i \pi b}{2}} c_2 x^{b+1} \, _2F_1\left (\frac {1}{4} \left (a+2 b-\sqrt {(a-1)^2-4 c}+1\right ),\frac {1}{4} \left (a+2 b+\sqrt {(a-1)^2-4 c}+1\right );\frac {b+3}{2};x^2\right ) \\ \end{align*}