Internal problem ID [9638]
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: 51.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, _Riccati]
Solve \begin {gather*} \boxed {x^{2} y^{\prime }-a \,x^{2} y^{2}-b x y-c \,x^{n}-s=0} \end {gather*}
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 291
dsolve(x^2*diff(y(x),x)=a*x^2*y(x)^2+b*x*y(x)+c*x^n+s,y(x), singsol=all)
\[ y \relax (x ) = -\frac {\left (\sqrt {-4 a s +b^{2}+2 b +1}\, c_{1}+b c_{1}+c_{1}\right ) \BesselY \left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{n}, \frac {2 \sqrt {a c}\, x^{\frac {n}{2}}}{n}\right )-2 \sqrt {a c}\, x^{\frac {n}{2}} \BesselY \left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}+n}{n}, \frac {2 \sqrt {a c}\, x^{\frac {n}{2}}}{n}\right ) c_{1}+\left (\sqrt {-4 a s +b^{2}+2 b +1}+b +1\right ) \BesselJ \left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{n}, \frac {2 \sqrt {a c}\, x^{\frac {n}{2}}}{n}\right )-2 \BesselJ \left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}+n}{n}, \frac {2 \sqrt {a c}\, x^{\frac {n}{2}}}{n}\right ) \sqrt {a c}\, x^{\frac {n}{2}}}{2 x a \left (\BesselY \left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{n}, \frac {2 \sqrt {a c}\, x^{\frac {n}{2}}}{n}\right ) c_{1}+\BesselJ \left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{n}, \frac {2 \sqrt {a c}\, x^{\frac {n}{2}}}{n}\right )\right )} \]
✓ Solution by Mathematica
Time used: 2.429 (sec). Leaf size: 2281
DSolve[x^2*y'[x]==a*x^2*y[x]^2+b*x*y[x]+c*x^n+s,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {a^{\frac {\sqrt {b^2+2 b-4 a s+1}}{n}} n^{\frac {2 \sqrt {n^2 \left (b^2+2 b-4 a s+1\right )}}{n^2}} \left (\sqrt {n^2 \left (b^2+2 b-4 a s+1\right )}-n \left (b+\sqrt {b^2+2 b-4 a s+1}+1\right )\right ) \left (x^n\right )^{\frac {\sqrt {b^2+2 b-4 a s+1}}{n}+\frac {1}{2}} J_{\frac {\sqrt {b^2+2 b-4 a s+1}}{n}}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^n}}{n}\right ) \Gamma \left (\frac {n+\sqrt {b^2+2 b-4 a s+1}}{n}\right ) c^{\frac {\sqrt {b^2+2 b-4 a s+1}}{n}}-a^{\frac {\sqrt {n^2 \left (b^2+2 b-4 a s+1\right )}}{n^2}} n^{\frac {2 \sqrt {b^2+2 b-4 a s+1}}{n}+1} \left (x^n\right )^{\frac {\sqrt {n^2 \left (b^2+2 b-4 a s+1\right )}}{n^2}+\frac {1}{2}} J_{-\frac {\sqrt {b^2+2 b-4 a s+1}}{n}}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^n}}{n}\right ) c_1 \Gamma \left (1-\frac {\sqrt {b^2+2 b-4 a s+1}}{n}\right ) c^{\frac {\sqrt {n^2 \left (b^2+2 b-4 a s+1\right )}}{n^2}}-a^{\frac {\sqrt {n^2 \left (b^2+2 b-4 a s+1\right )}}{n^2}} b n^{\frac {2 \sqrt {b^2+2 b-4 a s+1}}{n}+1} \left (x^n\right )^{\frac {\sqrt {n^2 \left (b^2+2 b-4 a s+1\right )}}{n^2}+\frac {1}{2}} J_{-\frac {\sqrt {b^2+2 b-4 a s+1}}{n}}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^n}}{n}\right ) c_1 \Gamma \left (1-\frac {\sqrt {b^2+2 b-4 a s+1}}{n}\right ) c^{\frac {\sqrt {n^2 \left (b^2+2 b-4 a s+1\right )}}{n^2}}+a^{\frac {\sqrt {n^2 \left (b^2+2 b-4 a s+1\right )}}{n^2}} n^{\frac {2 \sqrt {b^2+2 b-4 a s+1}}{n}+1} \sqrt {b^2+2 b-4 a s+1} \left (x^n\right )^{\frac {\sqrt {n^2 \left (b^2+2 b-4 a s+1\right )}}{n^2}+\frac {1}{2}} J_{-\frac {\sqrt {b^2+2 b-4 a s+1}}{n}}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^n}}{n}\right ) c_1 \Gamma \left (1-\frac {\sqrt {b^2+2 b-4 a s+1}}{n}\right ) c^{\frac {\sqrt {n^2 \left (b^2+2 b-4 a s+1\right )}}{n^2}}-a^{\frac {\sqrt {n^2 \left (b^2+2 b-4 a s+1\right )}}{n^2}} n^{\frac {2 \sqrt {b^2+2 b-4 a s+1}}{n}} \sqrt {n^2 \left (b^2+2 b-4 a s+1\right )} \left (x^n\right )^{\frac {\sqrt {n^2 \left (b^2+2 b-4 a s+1\right )}}{n^2}+\frac {1}{2}} J_{-\frac {\sqrt {b^2+2 b-4 a s+1}}{n}}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^n}}{n}\right ) c_1 \Gamma \left (1-\frac {\sqrt {b^2+2 b-4 a s+1}}{n}\right ) c^{\frac {\sqrt {n^2 \left (b^2+2 b-4 a s+1\right )}}{n^2}}+a^{\frac {\sqrt {b^2+2 b-4 a s+1}}{n}+\frac {1}{2}} n^{\frac {2 \sqrt {n^2 \left (b^2+2 b-4 a s+1\right )}}{n^2}+1} \left (x^n\right )^{\frac {n+\sqrt {b^2+2 b-4 a s+1}}{n}} J_{\frac {n+\sqrt {b^2+2 b-4 a s+1}}{n}}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^n}}{n}\right ) \Gamma \left (\frac {n+\sqrt {b^2+2 b-4 a s+1}}{n}\right ) c^{\frac {\sqrt {b^2+2 b-4 a s+1}}{n}+\frac {1}{2}}-a^{\frac {\sqrt {b^2+2 b-4 a s+1}}{n}+\frac {1}{2}} n^{\frac {2 \sqrt {n^2 \left (b^2+2 b-4 a s+1\right )}}{n^2}+1} \left (x^n\right )^{\frac {n+\sqrt {b^2+2 b-4 a s+1}}{n}} J_{\frac {\sqrt {b^2+2 b-4 a s+1}}{n}-1}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^n}}{n}\right ) \Gamma \left (\frac {n+\sqrt {b^2+2 b-4 a s+1}}{n}\right ) c^{\frac {\sqrt {b^2+2 b-4 a s+1}}{n}+\frac {1}{2}}-a^{\frac {\sqrt {n^2 \left (b^2+2 b-4 a s+1\right )}}{n^2}+\frac {1}{2}} n^{\frac {2 \sqrt {b^2+2 b-4 a s+1}}{n}+1} \left (x^n\right )^{\frac {\sqrt {n^2 \left (b^2+2 b-4 a s+1\right )}}{n^2}+1} J_{-\frac {n+\sqrt {b^2+2 b-4 a s+1}}{n}}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^n}}{n}\right ) c_1 \Gamma \left (1-\frac {\sqrt {b^2+2 b-4 a s+1}}{n}\right ) c^{\frac {\sqrt {n^2 \left (b^2+2 b-4 a s+1\right )}}{n^2}+\frac {1}{2}}+a^{\frac {\sqrt {n^2 \left (b^2+2 b-4 a s+1\right )}}{n^2}+\frac {1}{2}} n^{\frac {2 \sqrt {b^2+2 b-4 a s+1}}{n}+1} \left (x^n\right )^{\frac {\sqrt {n^2 \left (b^2+2 b-4 a s+1\right )}}{n^2}+1} J_{1-\frac {\sqrt {b^2+2 b-4 a s+1}}{n}}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^n}}{n}\right ) c_1 \Gamma \left (1-\frac {\sqrt {b^2+2 b-4 a s+1}}{n}\right ) c^{\frac {\sqrt {n^2 \left (b^2+2 b-4 a s+1\right )}}{n^2}+\frac {1}{2}}}{2 a n x \sqrt {x^n} \left (a^{\frac {\sqrt {b^2+2 b-4 a s+1}}{n}} c^{\frac {\sqrt {b^2+2 b-4 a s+1}}{n}} n^{\frac {2 \sqrt {n^2 \left (b^2+2 b-4 a s+1\right )}}{n^2}} J_{\frac {\sqrt {b^2+2 b-4 a s+1}}{n}}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^n}}{n}\right ) \Gamma \left (\frac {n+\sqrt {b^2+2 b-4 a s+1}}{n}\right ) \left (x^n\right )^{\frac {\sqrt {b^2+2 b-4 a s+1}}{n}}+a^{\frac {\sqrt {n^2 \left (b^2+2 b-4 a s+1\right )}}{n^2}} c^{\frac {\sqrt {n^2 \left (b^2+2 b-4 a s+1\right )}}{n^2}} n^{\frac {2 \sqrt {b^2+2 b-4 a s+1}}{n}} J_{-\frac {\sqrt {b^2+2 b-4 a s+1}}{n}}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^n}}{n}\right ) c_1 \Gamma \left (1-\frac {\sqrt {b^2+2 b-4 a s+1}}{n}\right ) \left (x^n\right )^{\frac {\sqrt {n^2 \left (b^2+2 b-4 a s+1\right )}}{n^2}}\right )} \\ y(x)\to \frac {\frac {\sqrt {a} \sqrt {c} \sqrt {x^n} \left (J_{1-\frac {\sqrt {b^2+2 b-4 a s+1}}{n}}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^n}}{n}\right )-J_{-\frac {n+\sqrt {b^2+2 b-4 a s+1}}{n}}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^n}}{n}\right )\right )}{J_{-\frac {\sqrt {b^2+2 b-4 a s+1}}{n}}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^n}}{n}\right )}-\frac {\sqrt {n^2 \left (-4 a s+b^2+2 b+1\right )}}{n}+\sqrt {-4 a s+b^2+2 b+1}-b-1}{2 a x} \\ \end{align*}