Internal problem ID [10992]
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: 158.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {\left (x^{2}-1\right ) y^{\prime \prime }+\left (1+2 a \right ) y^{\prime }-b \left (2 a +b \right ) y=0} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 112
dsolve((x^2-1)*diff(y(x),x$2)+(2*a+1)*diff(y(x),x)-b*(2*a+b)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} \operatorname {hypergeom}\left (\left [-\frac {1}{2}-\frac {\sqrt {8 a b +4 b^{2}+1}}{2}, \frac {\sqrt {8 a b +4 b^{2}+1}}{2}-\frac {1}{2}\right ], \left [-a -\frac {1}{2}\right ], \frac {x}{2}+\frac {1}{2}\right )+c_{2} \left (\frac {x}{2}+\frac {1}{2}\right )^{\frac {3}{2}+a} \operatorname {hypergeom}\left (\left [1-\frac {\sqrt {8 a b +4 b^{2}+1}}{2}+a , \frac {\sqrt {8 a b +4 b^{2}+1}}{2}+1+a \right ], \left [\frac {5}{2}+a \right ], \frac {x}{2}+\frac {1}{2}\right ) \]
✓ Solution by Mathematica
Time used: 0.304 (sec). Leaf size: 152
DSolve[(x^2-1)*y''[x]+(2*a+1)*y'[x]-b*(2*a+b)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to 2^{a-\frac {1}{2}} c_2 (x-1)^{\frac {1}{2}-a} \operatorname {Hypergeometric2F1}\left (-a-\frac {1}{2} \sqrt {4 b^2+8 a b+1},\frac {1}{2} \sqrt {4 b^2+8 a b+1}-a,\frac {3}{2}-a,\frac {1}{2}-\frac {x}{2}\right )+c_1 \operatorname {Hypergeometric2F1}\left (\frac {1}{2} \left (-\sqrt {4 b^2+8 a b+1}-1\right ),\frac {1}{2} \left (\sqrt {4 b^2+8 a b+1}-1\right ),a+\frac {1}{2},\frac {1-x}{2}\right ) \]