Internal problem ID [9629]
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: 42.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, _Riccati]
Solve \begin {gather*} \boxed {y^{\prime } x -a \,x^{n} y^{2}-b y-c \,x^{m}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.001 (sec). Leaf size: 172
dsolve(x*diff(y(x),x)=a*x^(n)*y(x)^2+b*y(x)+c*x^(m),y(x), singsol=all)
\[ y \relax (x ) = \frac {\left (\BesselY \left (-\frac {b -m}{n +m}, \frac {2 \sqrt {a c}\, x^{\frac {n}{2}+\frac {m}{2}}}{n +m}\right ) c_{1}+\BesselJ \left (-\frac {b -m}{n +m}, \frac {2 \sqrt {a c}\, x^{\frac {n}{2}+\frac {m}{2}}}{n +m}\right )\right ) x^{\frac {n}{2}+\frac {m}{2}} \sqrt {a c}\, x^{1-n}}{\left (\BesselY \left (-\frac {b +n}{n +m}, \frac {2 \sqrt {a c}\, x^{\frac {n}{2}+\frac {m}{2}}}{n +m}\right ) c_{1}+\BesselJ \left (-\frac {b +n}{n +m}, \frac {2 \sqrt {a c}\, x^{\frac {n}{2}+\frac {m}{2}}}{n +m}\right )\right ) a x} \]
✓ Solution by Mathematica
Time used: 1.308 (sec). Leaf size: 1578
DSolve[x*y'[x]==a*x^(n)*y[x]^2+b*y[x]+c*x^(m),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {x^{-n} \left (\sqrt {a} \sqrt {c} (m+n) x^{m+n} J_{\frac {m-b}{m+n}}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right ) c_1 \Gamma \left (\frac {m-b}{m+n}\right ) \left ((m+n)^2\right )^{\frac {b+n}{m+n}}-\sqrt {a} \sqrt {c} m x^{m+n} J_{-\frac {b+m+2 n}{m+n}}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right ) c_1 \Gamma \left (\frac {m-b}{m+n}\right ) \left ((m+n)^2\right )^{\frac {b+n}{m+n}}-\sqrt {a} \sqrt {c} n x^{m+n} J_{-\frac {b+m+2 n}{m+n}}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right ) c_1 \Gamma \left (\frac {m-b}{m+n}\right ) \left ((m+n)^2\right )^{\frac {b+n}{m+n}}-(b+n) \sqrt {x^{m+n}} J_{-\frac {b+n}{m+n}}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right ) c_1 \Gamma \left (\frac {m-b}{m+n}\right ) \left ((m+n)^2\right )^{\frac {2 b+m+3 n}{2 (m+n)}}-b (m+n)^{\frac {2 (b+n)}{m+n}} \sqrt {x^{m+n}} J_{\frac {b+n}{m+n}}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right ) \Gamma \left (\frac {b+m+2 n}{m+n}\right ) \sqrt {(m+n)^2}-n (m+n)^{\frac {2 (b+n)}{m+n}} \sqrt {x^{m+n}} J_{\frac {b+n}{m+n}}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right ) \Gamma \left (\frac {b+m+2 n}{m+n}\right ) \sqrt {(m+n)^2}-\sqrt {a} \sqrt {c} m (m+n)^{\frac {2 (b+n)}{m+n}} x^{m+n} J_{\frac {b-m}{m+n}}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right ) \Gamma \left (\frac {b+m+2 n}{m+n}\right )-\sqrt {a} \sqrt {c} n (m+n)^{\frac {2 (b+n)}{m+n}} x^{m+n} J_{\frac {b-m}{m+n}}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right ) \Gamma \left (\frac {b+m+2 n}{m+n}\right )+\sqrt {a} \sqrt {c} m (m+n)^{\frac {2 (b+n)}{m+n}} x^{m+n} J_{\frac {b+n}{m+n}+1}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right ) \Gamma \left (\frac {b+m+2 n}{m+n}\right )+\sqrt {a} \sqrt {c} n (m+n)^{\frac {2 (b+n)}{m+n}} x^{m+n} J_{\frac {b+n}{m+n}+1}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right ) \Gamma \left (\frac {b+m+2 n}{m+n}\right )\right )}{2 a \sqrt {(m+n)^2} \sqrt {x^{m+n}} \left (J_{-\frac {b+n}{m+n}}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right ) c_1 \Gamma \left (\frac {m-b}{m+n}\right ) \left ((m+n)^2\right )^{\frac {b+n}{m+n}}+(m+n)^{\frac {2 (b+n)}{m+n}} J_{\frac {b+n}{m+n}}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right ) \Gamma \left (\frac {b+m+2 n}{m+n}\right )\right )} \\ y(x)\to \frac {x^{-n} \left (\sqrt {a} \sqrt {c} (m+n) \sqrt {x^{m+n}} J_{\frac {m-b}{m+n}}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right )-(b+n) \sqrt {(m+n)^2} J_{-\frac {b+n}{m+n}}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right )-\sqrt {a} \sqrt {c} (m+n) \sqrt {x^{m+n}} J_{-\frac {b+m+2 n}{m+n}}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right )\right )}{2 a \sqrt {(m+n)^2} J_{-\frac {b+n}{m+n}}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right )} \\ y(x)\to \frac {x^{-n} \left (\sqrt {a} \sqrt {c} (m+n) \sqrt {x^{m+n}} J_{\frac {m-b}{m+n}}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right )-(b+n) \sqrt {(m+n)^2} J_{-\frac {b+n}{m+n}}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right )-\sqrt {a} \sqrt {c} (m+n) \sqrt {x^{m+n}} J_{-\frac {b+m+2 n}{m+n}}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right )\right )}{2 a \sqrt {(m+n)^2} J_{-\frac {b+n}{m+n}}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right )} \\ \end{align*}