Internal problem ID [10996]
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]]
\[ \boxed {x \left (x +a \right ) y^{\prime \prime }+\left (b x +c \right ) y^{\prime }+y d=0} \]
✓ Solution by Maple
Time used: 0.109 (sec). Leaf size: 230
dsolve(x*(x+a)*diff(y(x),x$2)+(b*x+c)*diff(y(x),x)+d*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{2} \left (\operatorname {csgn}\left (a \right ) a +a +2 x \right )^{-\frac {\left (\left (b -2\right ) \operatorname {csgn}\left (a \right ) a +a b -2 c \right ) \operatorname {csgn}\left (a \right )}{2 a}} \operatorname {hypergeom}\left (\left [\frac {\operatorname {csgn}\left (a \right ) \left (\operatorname {csgn}\left (a \right ) a +\sqrt {b^{2}-2 b -4 d +1}\, \operatorname {csgn}\left (a \right ) a -a b +2 c \right )}{2 a}, -\frac {\operatorname {csgn}\left (a \right ) \left (\sqrt {b^{2}-2 b -4 d +1}\, \operatorname {csgn}\left (a \right ) a -\operatorname {csgn}\left (a \right ) a +a b -2 c \right )}{2 a}\right ], \left [-\frac {\operatorname {csgn}\left (a \right ) \left (\left (b -4\right ) \operatorname {csgn}\left (a \right ) a +a b -2 c \right )}{2 a}\right ], \frac {\operatorname {csgn}\left (a \right ) \left (\operatorname {csgn}\left (a \right ) a +a +2 x \right )}{2 a}\right )+c_{1} \operatorname {hypergeom}\left (\left [-\frac {1}{2}+\frac {b}{2}-\frac {\sqrt {b^{2}-2 b -4 d +1}}{2}, -\frac {1}{2}+\frac {b}{2}+\frac {\sqrt {b^{2}-2 b -4 d +1}}{2}\right ], \left [\frac {\left (b \,\operatorname {csgn}\left (a \right ) a +a b -2 c \right ) \operatorname {csgn}\left (a \right )}{2 a}\right ], \frac {\operatorname {csgn}\left (a \right ) \left (\operatorname {csgn}\left (a \right ) a +a +2 x \right )}{2 a}\right ) \]
✓ 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]
\[ y(x)\to c_2 a^{\frac {c}{a}-1} x^{1-\frac {c}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} \left (b-\frac {2 c}{a}+\sqrt {b^2-2 b-4 d+1}+1\right ),\frac {b a-\sqrt {b^2-2 b-4 d+1} a+a-2 c}{2 a},2-\frac {c}{a},-\frac {x}{a}\right )+c_1 \operatorname {Hypergeometric2F1}\left (\frac {1}{2} \left (b-\sqrt {b^2-2 b-4 d+1}-1\right ),\frac {1}{2} \left (b+\sqrt {b^2-2 b-4 d+1}-1\right ),\frac {c}{a},-\frac {x}{a}\right ) \]