Internal problem ID [10397]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 1, section 1.2. Riccati Equation. 1.2.2. Equations Containing Power
Functions
Problem number: 67.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, _Riccati]
\[ \boxed {x \left (x^{2}+a \right ) \left (y^{\prime }+\lambda y^{2}\right )+\left (b \,x^{2}+c \right ) y=-s x} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 613
dsolve(x*(x^2+a)*(diff(y(x),x)+lambda*y(x)^2)+(b*x^2+c)*y(x)+s*x=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\left (a -\frac {c}{3}\right ) \left (\left (-\lambda s +b -2\right ) a^{2}+c \left (b -3\right ) a -c^{2}\right ) \left (a \,x^{\frac {a +c}{a}}+x^{\frac {3 a +c}{a}}\right ) c_{1} \operatorname {hypergeom}\left (\left [-\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}\, a +a b -7 a -2 c}{4 a}, \frac {\sqrt {b^{2}-4 \lambda s -2 b +1}\, a +\left (-b +7\right ) a +2 c}{4 a}\right ], \left [\frac {3 a +c}{2 a}\right ], -\frac {x^{2}}{a}\right )+\left (-\left (a -\frac {c}{3}\right ) \left (\left (b -2\right ) a -c \right ) c_{1} x^{\frac {a +c}{a}} \operatorname {hypergeom}\left (\left [-\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}\, a +a b -3 a -2 c}{4 a}, \frac {\sqrt {b^{2}-4 \lambda s -2 b +1}\, a +\left (-b +3\right ) a +2 c}{4 a}\right ], \left [\frac {a +c}{2 a}\right ], -\frac {x^{2}}{a}\right )+\left (\frac {2 \left (x^{2}+a \right ) x^{2} \left (-\frac {\lambda s}{2}+b -3\right ) \operatorname {hypergeom}\left (\left [-\frac {b}{4}+\frac {9}{4}+\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}}{4}, -\frac {b}{4}+\frac {9}{4}-\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}}{4}\right ], \left [\frac {5 a -c}{2 a}\right ], -\frac {x^{2}}{a}\right )}{3}+\left (a -\frac {c}{3}\right ) \left (a -c +\left (-b +3\right ) x^{2}\right ) \operatorname {hypergeom}\left (\left [-\frac {b}{4}+\frac {5}{4}+\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}}{4}, -\frac {b}{4}+\frac {5}{4}-\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}}{4}\right ], \left [\frac {3 a -c}{2 a}\right ], -\frac {x^{2}}{a}\right )\right ) a \right ) \left (a +c \right ) a}{\left (a -\frac {c}{3}\right ) \left (x \operatorname {hypergeom}\left (\left [-\frac {b}{4}+\frac {5}{4}+\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}}{4}, -\frac {b}{4}+\frac {5}{4}-\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}}{4}\right ], \left [\frac {3 a -c}{2 a}\right ], -\frac {x^{2}}{a}\right )+\operatorname {hypergeom}\left (\left [-\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}\, a +a b -3 a -2 c}{4 a}, \frac {\sqrt {b^{2}-4 \lambda s -2 b +1}\, a +\left (-b +3\right ) a +2 c}{4 a}\right ], \left [\frac {a +c}{2 a}\right ], -\frac {x^{2}}{a}\right ) c_{1} x^{\frac {c}{a}}\right ) \left (x^{2}+a \right ) \left (a +c \right ) a^{2} \lambda } \]
✓ Solution by Mathematica
Time used: 2.874 (sec). Leaf size: 862
DSolve[x*(x^2+a)*(y'[x]+\[Lambda]*y[x]^2)+(b*x^2+c)*y[x]+s*x==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {a^{\frac {1}{2} \left (\frac {c}{a}-3\right )} (a-c) x^{-\frac {c}{a}} \operatorname {Hypergeometric2F1}\left (\frac {b a-\sqrt {b^2-2 b-4 s \lambda +1} a+a-2 c}{4 a},\frac {a \left (b+\sqrt {b^2-2 b-4 s \lambda +1}+1\right )-2 c}{4 a},\frac {3}{2}-\frac {c}{2 a},-\frac {x^2}{a}\right )+\frac {a^{\frac {1}{2} \left (\frac {c}{a}-5\right )} x^{2-\frac {c}{a}} \left (a \left (\sqrt {b^2-2 b-4 \lambda s+1}-b-1\right )+2 c\right ) \left (a \left (\sqrt {b^2-2 b-4 \lambda s+1}+b+1\right )-2 c\right ) \operatorname {Hypergeometric2F1}\left (\frac {a \left (b-\sqrt {b^2-2 b-4 s \lambda +1}+5\right )-2 c}{4 a},\frac {a \left (b+\sqrt {b^2-2 b-4 s \lambda +1}+5\right )-2 c}{4 a},\frac {5}{2}-\frac {c}{2 a},-\frac {x^2}{a}\right )}{4 (3 a-c)}-\frac {c_1 \lambda s x \operatorname {Hypergeometric2F1}\left (\frac {1}{4} \left (b-\sqrt {b^2-2 b-4 s \lambda +1}+3\right ),\frac {1}{4} \left (b+\sqrt {b^2-2 b-4 s \lambda +1}+3\right ),\frac {1}{2} \left (\frac {c}{a}+3\right ),-\frac {x^2}{a}\right )}{a+c}}{\lambda a^{\frac {1}{2} \left (\frac {c}{a}-1\right )} x^{1-\frac {c}{a}} \operatorname {Hypergeometric2F1}\left (\frac {b a-\sqrt {b^2-2 b-4 s \lambda +1} a+a-2 c}{4 a},\frac {a \left (b+\sqrt {b^2-2 b-4 s \lambda +1}+1\right )-2 c}{4 a},\frac {3}{2}-\frac {c}{2 a},-\frac {x^2}{a}\right )+c_1 \lambda \operatorname {Hypergeometric2F1}\left (\frac {1}{4} \left (b-\sqrt {b^2-2 b-4 s \lambda +1}-1\right ),\frac {1}{4} \left (b+\sqrt {b^2-2 b-4 s \lambda +1}-1\right ),\frac {a+c}{2 a},-\frac {x^2}{a}\right )} \\ y(x)\to -\frac {s x \operatorname {Hypergeometric2F1}\left (\frac {1}{4} \left (b-\sqrt {b^2-2 b-4 s \lambda +1}+3\right ),\frac {1}{4} \left (b+\sqrt {b^2-2 b-4 s \lambda +1}+3\right ),\frac {1}{2} \left (\frac {c}{a}+3\right ),-\frac {x^2}{a}\right )}{(a+c) \operatorname {Hypergeometric2F1}\left (\frac {1}{4} \left (b-\sqrt {b^2-2 b-4 s \lambda +1}-1\right ),\frac {1}{4} \left (b+\sqrt {b^2-2 b-4 s \lambda +1}-1\right ),\frac {a+c}{2 a},-\frac {x^2}{a}\right )} \\ y(x)\to -\frac {s x \operatorname {Hypergeometric2F1}\left (\frac {1}{4} \left (b-\sqrt {b^2-2 b-4 s \lambda +1}+3\right ),\frac {1}{4} \left (b+\sqrt {b^2-2 b-4 s \lambda +1}+3\right ),\frac {1}{2} \left (\frac {c}{a}+3\right ),-\frac {x^2}{a}\right )}{(a+c) \operatorname {Hypergeometric2F1}\left (\frac {1}{4} \left (b-\sqrt {b^2-2 b-4 s \lambda +1}-1\right ),\frac {1}{4} \left (b+\sqrt {b^2-2 b-4 s \lambda +1}-1\right ),\frac {a+c}{2 a},-\frac {x^2}{a}\right )} \\ \end{align*}