Internal problem ID [9608]
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: 21.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
Solve \begin {gather*} \boxed {x^{n +1} y^{\prime }-x^{2 n} a y^{2}-c \,x^{m}-d=0} \end {gather*}
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 264
dsolve(x^(n+1)*diff(y(x),x)=a*x^(2*n)*y(x)^2+c*x^m+d,y(x), singsol=all)
\[ y \relax (x ) = -\frac {\left (\left (\sqrt {-4 a d +n^{2}}\, c_{1}+c_{1} n \right ) \BesselY \left (\frac {\sqrt {-4 a d +n^{2}}}{m}, \frac {2 \sqrt {a c}\, x^{\frac {m}{2}}}{m}\right )-2 x^{\frac {m}{2}} \BesselY \left (\frac {\sqrt {-4 a d +n^{2}}+m}{m}, \frac {2 \sqrt {a c}\, x^{\frac {m}{2}}}{m}\right ) \sqrt {a c}\, c_{1}+\left (\sqrt {-4 a d +n^{2}}+n \right ) \BesselJ \left (\frac {\sqrt {-4 a d +n^{2}}}{m}, \frac {2 \sqrt {a c}\, x^{\frac {m}{2}}}{m}\right )-2 \BesselJ \left (\frac {\sqrt {-4 a d +n^{2}}+m}{m}, \frac {2 \sqrt {a c}\, x^{\frac {m}{2}}}{m}\right ) \sqrt {a c}\, x^{\frac {m}{2}}\right ) x^{1-n}}{2 x a \left (\BesselY \left (\frac {\sqrt {-4 a d +n^{2}}}{m}, \frac {2 \sqrt {a c}\, x^{\frac {m}{2}}}{m}\right ) c_{1}+\BesselJ \left (\frac {\sqrt {-4 a d +n^{2}}}{m}, \frac {2 \sqrt {a c}\, x^{\frac {m}{2}}}{m}\right )\right )} \]
✓ Solution by Mathematica
Time used: 1.871 (sec). Leaf size: 1890
DSolve[x^(n+1)*y'[x]==a*x^(2*n)*y[x]^2+c*x^m+d,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {x^{-n} \left (a^{\frac {\sqrt {n^2-4 a d}}{m}} m^{\frac {2 \sqrt {m^2 \left (n^2-4 a d\right )}}{m^2}} \left (\sqrt {m^2 \left (n^2-4 a d\right )}-m \left (n+\sqrt {n^2-4 a d}\right )\right ) \left (x^m\right )^{\frac {\sqrt {n^2-4 a d}}{m}+\frac {1}{2}} J_{\frac {\sqrt {m^2 \left (n^2-4 a d\right )}}{m^2}}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^m}}{m}\right ) \Gamma \left (\frac {m+\sqrt {n^2-4 a d}}{m}\right ) c^{\frac {\sqrt {n^2-4 a d}}{m}}-a^{\frac {\sqrt {m^2 \left (n^2-4 a d\right )}}{m^2}} m^{\frac {2 \sqrt {n^2-4 a d}}{m}+1} n \left (x^m\right )^{\frac {\sqrt {m^2 \left (n^2-4 a d\right )}}{m^2}+\frac {1}{2}} J_{-\frac {\sqrt {m^2 \left (n^2-4 a d\right )}}{m^2}}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^m}}{m}\right ) c_1 \Gamma \left (1-\frac {\sqrt {n^2-4 a d}}{m}\right ) c^{\frac {\sqrt {m^2 \left (n^2-4 a d\right )}}{m^2}}+a^{\frac {\sqrt {m^2 \left (n^2-4 a d\right )}}{m^2}} m^{\frac {2 \sqrt {n^2-4 a d}}{m}+1} \sqrt {n^2-4 a d} \left (x^m\right )^{\frac {\sqrt {m^2 \left (n^2-4 a d\right )}}{m^2}+\frac {1}{2}} J_{-\frac {\sqrt {m^2 \left (n^2-4 a d\right )}}{m^2}}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^m}}{m}\right ) c_1 \Gamma \left (1-\frac {\sqrt {n^2-4 a d}}{m}\right ) c^{\frac {\sqrt {m^2 \left (n^2-4 a d\right )}}{m^2}}-a^{\frac {\sqrt {m^2 \left (n^2-4 a d\right )}}{m^2}} m^{\frac {2 \sqrt {n^2-4 a d}}{m}} \sqrt {m^2 \left (n^2-4 a d\right )} \left (x^m\right )^{\frac {\sqrt {m^2 \left (n^2-4 a d\right )}}{m^2}+\frac {1}{2}} J_{-\frac {\sqrt {m^2 \left (n^2-4 a d\right )}}{m^2}}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^m}}{m}\right ) c_1 \Gamma \left (1-\frac {\sqrt {n^2-4 a d}}{m}\right ) c^{\frac {\sqrt {m^2 \left (n^2-4 a d\right )}}{m^2}}+a^{\frac {\sqrt {n^2-4 a d}}{m}+\frac {1}{2}} m^{\frac {m^2+2 \sqrt {m^2 \left (n^2-4 a d\right )}}{m^2}} \left (x^m\right )^{\frac {m+\sqrt {n^2-4 a d}}{m}} J_{\frac {m^2+\sqrt {m^2 \left (n^2-4 a d\right )}}{m^2}}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^m}}{m}\right ) \Gamma \left (\frac {m+\sqrt {n^2-4 a d}}{m}\right ) c^{\frac {\sqrt {n^2-4 a d}}{m}+\frac {1}{2}}-a^{\frac {\sqrt {n^2-4 a d}}{m}+\frac {1}{2}} m^{\frac {m^2+2 \sqrt {m^2 \left (n^2-4 a d\right )}}{m^2}} \left (x^m\right )^{\frac {m+\sqrt {n^2-4 a d}}{m}} J_{\frac {\sqrt {m^2 \left (n^2-4 a d\right )}}{m^2}-1}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^m}}{m}\right ) \Gamma \left (\frac {m+\sqrt {n^2-4 a d}}{m}\right ) c^{\frac {\sqrt {n^2-4 a d}}{m}+\frac {1}{2}}-a^{\frac {\sqrt {m^2 \left (n^2-4 a d\right )}}{m^2}+\frac {1}{2}} m^{\frac {2 \sqrt {n^2-4 a d}}{m}+1} \left (x^m\right )^{\frac {m^2+\sqrt {m^2 \left (n^2-4 a d\right )}}{m^2}} J_{-\frac {\sqrt {m^2 \left (n^2-4 a d\right )}}{m^2}-1}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^m}}{m}\right ) c_1 \Gamma \left (1-\frac {\sqrt {n^2-4 a d}}{m}\right ) c^{\frac {\sqrt {m^2 \left (n^2-4 a d\right )}}{m^2}+\frac {1}{2}}+a^{\frac {\sqrt {m^2 \left (n^2-4 a d\right )}}{m^2}+\frac {1}{2}} m^{\frac {2 \sqrt {n^2-4 a d}}{m}+1} \left (x^m\right )^{\frac {m^2+\sqrt {m^2 \left (n^2-4 a d\right )}}{m^2}} J_{1-\frac {\sqrt {m^2 \left (n^2-4 a d\right )}}{m^2}}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^m}}{m}\right ) c_1 \Gamma \left (1-\frac {\sqrt {n^2-4 a d}}{m}\right ) c^{\frac {\sqrt {m^2 \left (n^2-4 a d\right )}}{m^2}+\frac {1}{2}}\right )}{2 a m \sqrt {x^m} \left (a^{\frac {\sqrt {n^2-4 a d}}{m}} c^{\frac {\sqrt {n^2-4 a d}}{m}} m^{\frac {2 \sqrt {m^2 \left (n^2-4 a d\right )}}{m^2}} J_{\frac {\sqrt {m^2 \left (n^2-4 a d\right )}}{m^2}}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^m}}{m}\right ) \Gamma \left (\frac {m+\sqrt {n^2-4 a d}}{m}\right ) \left (x^m\right )^{\frac {\sqrt {n^2-4 a d}}{m}}+a^{\frac {\sqrt {m^2 \left (n^2-4 a d\right )}}{m^2}} c^{\frac {\sqrt {m^2 \left (n^2-4 a d\right )}}{m^2}} m^{\frac {2 \sqrt {n^2-4 a d}}{m}} J_{-\frac {\sqrt {m^2 \left (n^2-4 a d\right )}}{m^2}}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^m}}{m}\right ) c_1 \Gamma \left (1-\frac {\sqrt {n^2-4 a d}}{m}\right ) \left (x^m\right )^{\frac {\sqrt {m^2 \left (n^2-4 a d\right )}}{m^2}}\right )} \\ y(x)\to \frac {x^{-n} \left (\frac {\sqrt {a} \sqrt {c} \sqrt {x^m} \left (J_{1-\frac {\sqrt {m^2 \left (n^2-4 a d\right )}}{m^2}}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^m}}{m}\right )-J_{-\frac {\sqrt {m^2 \left (n^2-4 a d\right )}}{m^2}-1}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^m}}{m}\right )\right )}{J_{-\frac {\sqrt {m^2 \left (n^2-4 a d\right )}}{m^2}}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^m}}{m}\right )}-\frac {\sqrt {m^2 \left (n^2-4 a d\right )}}{m}+\sqrt {n^2-4 a d}-n\right )}{2 a} \\ \end{align*}