Internal problem ID [10993]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-5 Equation of form
\((a x^2+b x+c) y''+f(x)y'+g(x)y=0\)
Problem number: 159.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Gegenbauer]
\[ \boxed {\left (-x^{2}+1\right ) y^{\prime \prime }+\left (2 a -3\right ) x y^{\prime }+\left (1+n \right ) \left (n +2 a -1\right ) y=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 49
dsolve((1-x^2)*diff(y(x),x$2)+(2*a-3)*x*diff(y(x),x)+(n+1)*(n+2*a-1)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} \left (x^{2}-1\right )^{-\frac {1}{4}+\frac {a}{2}} \operatorname {LegendreP}\left (n +a -\frac {1}{2}, -\frac {1}{2}+a , x\right )+c_{2} \left (x^{2}-1\right )^{-\frac {1}{4}+\frac {a}{2}} \operatorname {LegendreQ}\left (n +a -\frac {1}{2}, -\frac {1}{2}+a , x\right ) \]
✓ Solution by Mathematica
Time used: 0.325 (sec). Leaf size: 158
DSolve[(1-x^2)*y''[x]+(2*a-3)*y'[x]+(n+1)*(n+2*a-1)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to 2^{\frac {1}{2}-a} c_2 (x-1)^{a-\frac {1}{2}} \operatorname {Hypergeometric2F1}\left (a-\frac {1}{2} \sqrt {4 n^2+8 a (n+1)-3}-1,a+\frac {1}{2} \sqrt {4 n^2+8 a (n+1)-3}-1,a+\frac {1}{2},\frac {1-x}{2}\right )+c_1 \operatorname {Hypergeometric2F1}\left (\frac {1}{2} \left (-\sqrt {4 n^2+8 a (n+1)-3}-1\right ),\frac {1}{2} \left (\sqrt {4 n^2+8 a (n+1)-3}-1\right ),\frac {3}{2}-a,\frac {1-x}{2}\right ) \]