Internal problem ID [11006]
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: 172.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x \left (x +a \right ) y^{\prime \prime }+\left (b x +c \right ) y^{\prime }+d y=0} \]
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 249
dsolve(x*(x+a)*diff(y(x),x$2)+(b*x+c)*diff(y(x),x)+d*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} \operatorname {hypergeom}\left (\left [-\frac {1}{2}+\frac {b}{2}-\frac {\sqrt {b^{2}-2 b -4 d +1}}{2}, -\frac {1}{2}+\frac {b}{2}+\frac {\sqrt {b^{2}-2 b -4 d +1}}{2}\right ], \left [\frac {b \sqrt {a^{2}}+a b -2 c}{2 \sqrt {a^{2}}}\right ], \frac {\sqrt {a^{2}}+a +2 x}{2 \sqrt {a^{2}}}\right )+c_{2} \left (\sqrt {a^{2}}+a +2 x \right )^{-\frac {\left (\left (b -2\right ) \sqrt {a^{2}}+a b -2 c \right ) \sqrt {a^{2}}}{2 a^{2}}} \operatorname {hypergeom}\left (\left [-\frac {\sqrt {b^{2}-2 b -4 d +1}\, \sqrt {a^{2}}-\sqrt {a^{2}}+a b -2 c}{2 \sqrt {a^{2}}}, \frac {\sqrt {a^{2}}+\sqrt {b^{2}-2 b -4 d +1}\, \sqrt {a^{2}}-a b +2 c}{2 \sqrt {a^{2}}}\right ], \left [-\frac {\left (b -4\right ) \sqrt {a^{2}}+a b -2 c}{2 \sqrt {a^{2}}}\right ], \frac {\sqrt {a^{2}}+a +2 x}{2 \sqrt {a^{2}}}\right ) \]
✓ Solution by Mathematica
Time used: 0.423 (sec). Leaf size: 165
DSolve[x*(x+a)*y''[x]+(b*x+c)*y'[x]+d*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_2 a^{\frac {c}{a}-1} x^{1-\frac {c}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} \left (b-\frac {2 c}{a}+\sqrt {b^2-2 b-4 d+1}+1\right ),\frac {b a-\sqrt {b^2-2 b-4 d+1} a+a-2 c}{2 a},2-\frac {c}{a},-\frac {x}{a}\right )+c_1 \operatorname {Hypergeometric2F1}\left (\frac {1}{2} \left (b-\sqrt {b^2-2 b-4 d+1}-1\right ),\frac {1}{2} \left (b+\sqrt {b^2-2 b-4 d+1}-1\right ),\frac {c}{a},-\frac {x}{a}\right ) \]