Internal problem ID [9691]
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]]
\[ \boxed {y^{\prime \prime }+\frac {\left (a \,x^{2}+a -2\right ) y^{\prime }}{x \left (x^{2}-1\right )}+\frac {b y}{x^{2}}=0} \]
✓ Solution by Maple
Time used: 0.172 (sec). Leaf size: 253
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)
\[ y \left (x \right ) = x^{\frac {a}{2}-\frac {1}{2}} \left (-x^{2}+1\right )^{\frac {1}{2}+\frac {a}{2}} \left (x^{2}-1\right )^{-a} \left (x^{\frac {\sqrt {a^{2}-2 a -4 b +1}}{2}} \left (x^{2}\right )^{-\frac {\sqrt {a^{2}-2 a -4 b +1}}{4}} \operatorname {LegendreP}\left (\frac {a}{2}-\frac {3}{2}, -\frac {\sqrt {a^{2}-2 a -4 b +1}}{2}, \frac {-x^{2}-1}{x^{2}-1}\right ) c_{1} \Gamma \left (1+\frac {\sqrt {a^{2}-2 a -4 b +1}}{2}\right )+\frac {\csc \left (\frac {\pi \sqrt {a^{2}-2 a -4 b +1}}{2}\right ) x^{-\frac {\sqrt {a^{2}-2 a -4 b +1}}{2}} \left (x^{2}\right )^{\frac {\sqrt {a^{2}-2 a -4 b +1}}{4}} \operatorname {LegendreP}\left (\frac {a}{2}-\frac {3}{2}, \frac {\sqrt {a^{2}-2 a -4 b +1}}{2}, \frac {-x^{2}-1}{x^{2}-1}\right ) \sqrt {a^{2}-2 a -4 b +1}\, \pi c_{2}}{2 \Gamma \left (1+\frac {\sqrt {a^{2}-2 a -4 b +1}}{2}\right )}\right ) \]
✓ Solution by Mathematica
Time used: 0.733 (sec). Leaf size: 212
DSolve[y''[x] == -((b*y[x])/x^2) - ((-2 + a + a*x^2)*y'[x])/(x*(-1 + x^2)),y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to -(-1)^{\frac {1}{4} \left (-\sqrt {a^2-2 a-4 b+1}+a+3\right )} x^{\frac {1}{2} \left (-\sqrt {a^2-2 a-4 b+1}+a-1\right )} \left (c_1 \operatorname {Hypergeometric2F1}\left (\frac {a-1}{2},\frac {1}{2} \left (a-\sqrt {a^2-2 a-4 b+1}-1\right ),1-\frac {1}{2} \sqrt {a^2-2 a-4 b+1},x^2\right )+c_2 i^{\sqrt {a^2-2 a-4 b+1}} x^{\sqrt {a^2-2 a-4 b+1}} \operatorname {Hypergeometric2F1}\left (\frac {a-1}{2},\frac {1}{2} \left (a+\sqrt {a^2-2 a-4 b+1}-1\right ),\frac {1}{2} \left (\sqrt {a^2-2 a-4 b+1}+2\right ),x^2\right )\right ) \]