Internal problem ID [8942]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1363.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+\frac {\left (a \,x^{2}+a -2\right ) y^{\prime }}{x \left (x^{2}-1\right )}+\frac {b y}{x^{2}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.025 (sec). Leaf size: 171
dsolve(diff(diff(y(x),x),x) = -1/x*(a*x^2+a-2)/(x^2-1)*diff(y(x),x)-b/x^2*y(x),y(x), singsol=all)
\[ y \relax (x ) = c_{1} x^{\frac {a}{2}-\frac {1}{2}+\frac {\sqrt {a^{2}-2 a -4 b +1}}{2}} \left (x^{2}-1\right )^{2-a} \hypergeom \left (\left [-\frac {a}{2}+\frac {3}{2}, -\frac {a}{2}+\frac {3}{2}+\frac {\sqrt {a^{2}-2 a -4 b +1}}{2}\right ], \left [1+\frac {\sqrt {a^{2}-2 a -4 b +1}}{2}\right ], x^{2}\right )+c_{2} x^{\frac {a}{2}-\frac {1}{2}-\frac {\sqrt {a^{2}-2 a -4 b +1}}{2}} \left (x^{2}-1\right )^{2-a} \hypergeom \left (\left [-\frac {a}{2}+\frac {3}{2}, -\frac {a}{2}+\frac {3}{2}-\frac {\sqrt {a^{2}-2 a -4 b +1}}{2}\right ], \left [1-\frac {\sqrt {a^{2}-2 a -4 b +1}}{2}\right ], x^{2}\right ) \]
✓ Solution by Mathematica
Time used: 0.377 (sec). Leaf size: 201
DSolve[y''[x] == -((b*y[x])/x^2) - ((-2 + a + a*x^2)*y'[x])/(x*(-1 + x^2)),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -(-1)^{\frac {1}{4} \left (-\sqrt {(a-1)^2-4 b}+a+3\right )} x^{\frac {1}{2} \left (-\sqrt {(a-1)^2-4 b}+a-1\right )} \left (c_1 \, _2F_1\left (\frac {a-1}{2},\frac {1}{2} \left (a-\sqrt {(a-1)^2-4 b}-1\right );1-\frac {1}{2} \sqrt {(a-1)^2-4 b};x^2\right )+c_2 e^{\frac {1}{2} i \pi \sqrt {(a-1)^2-4 b}} x^{\sqrt {(a-1)^2-4 b}} \, _2F_1\left (\frac {a-1}{2},\frac {1}{2} \left (a+\sqrt {(a-1)^2-4 b}-1\right );\frac {1}{2} \left (\sqrt {(a-1)^2-4 b}+2\right );x^2\right )\right ) \\ \end{align*}