Internal problem ID [11002]
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: 168.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {\left (-x^{2}+1\right ) y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+y c=0} \]
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 134
dsolve((1-x^2)*diff(y(x),x$2)+(a*x+b)*diff(y(x),x)+c*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} \operatorname {hypergeom}\left (\left [-\frac {1}{2}-\frac {a}{2}-\frac {\sqrt {a^{2}+2 a +4 c +1}}{2}, -\frac {1}{2}-\frac {a}{2}+\frac {\sqrt {a^{2}+2 a +4 c +1}}{2}\right ], \left [-\frac {a}{2}+\frac {b}{2}\right ], \frac {x}{2}+\frac {1}{2}\right )+c_{2} \left (\frac {x}{2}+\frac {1}{2}\right )^{1+\frac {a}{2}-\frac {b}{2}} \operatorname {hypergeom}\left (\left [\frac {1}{2}-\frac {\sqrt {a^{2}+2 a +4 c +1}}{2}-\frac {b}{2}, \frac {1}{2}+\frac {\sqrt {a^{2}+2 a +4 c +1}}{2}-\frac {b}{2}\right ], \left [2+\frac {a}{2}-\frac {b}{2}\right ], \frac {x}{2}+\frac {1}{2}\right ) \]
✓ Solution by Mathematica
Time used: 0.317 (sec). Leaf size: 184
DSolve[(1-x^2)*y''[x]+(a*x+b)*y'[x]+c*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to 2^{\frac {1}{2} (-a-b-2)} \left (c_2 (x-1)^{\frac {1}{2} (a+b+2)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} \left (b-\sqrt {a^2+2 a+4 c+1}+1\right ),\frac {1}{2} \left (b+\sqrt {a^2+2 a+4 c+1}+1\right ),\frac {1}{2} (a+b+4),\frac {1-x}{2}\right )+c_1 2^{\frac {1}{2} (a+b+2)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} \left (-a-\sqrt {a^2+2 a+4 c+1}-1\right ),\frac {1}{2} \left (-a+\sqrt {a^2+2 a+4 c+1}-1\right ),\frac {1}{2} (-a-b),\frac {1-x}{2}\right )\right ) \]