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: 0.0 (sec). Leaf size: 299
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)
\[ y \relax (x ) = \frac {\left (-\sqrt {-4 a s +b^{2}+2 b +1}\, c_{1}-c_{1} b -c_{1}\right ) \BesselY \left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{n}, \frac {2 \sqrt {c a}\, x^{\frac {n}{2}}}{n}\right )+2 x^{\frac {n}{2}} \sqrt {c a}\, \BesselY \left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}+n}{n}, \frac {2 \sqrt {c a}\, x^{\frac {n}{2}}}{n}\right ) c_{1}+\left (-\sqrt {-4 a s +b^{2}+2 b +1}-b -1\right ) \BesselJ \left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{n}, \frac {2 \sqrt {c a}\, x^{\frac {n}{2}}}{n}\right )+2 \BesselJ \left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}+n}{n}, \frac {2 \sqrt {c a}\, x^{\frac {n}{2}}}{n}\right ) \sqrt {c a}\, x^{\frac {n}{2}}}{2 x a \left (\BesselY \left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{n}, \frac {2 \sqrt {c a}\, x^{\frac {n}{2}}}{n}\right ) c_{1}+\BesselJ \left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{n}, \frac {2 \sqrt {c a}\, x^{\frac {n}{2}}}{n}\right )\right )} \]
✓ Solution by Mathematica
Time used: 1.33 (sec). Leaf size: 1105
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 {\left (\frac {\sqrt {a} \sqrt {c} \sqrt {x^n}}{n}\right )^{-\frac {\sqrt {(b+1)^2-4 a s}}{n}} \left (a^{\frac {\sqrt {(b+1)^2-4 a s}}{n}} c^{\frac {\sqrt {(b+1)^2-4 a s}}{n}} n^{\frac {2 \sqrt {n^2 \left ((b+1)^2-4 a s\right )}}{n^2}} \left (\frac {\sqrt {a} \sqrt {c} \sqrt {x^n}}{n}\right )^{\frac {2 \sqrt {(b+1)^2-4 a s}}{n}} \Gamma \left (\frac {\sqrt {(b+1)^2-4 a s}}{n}\right ) \left (2 a c \sqrt {(b+1)^2-4 a s} x^n \, _0\tilde {F}_1\left (;\frac {\sqrt {(b+1)^2-4 a s}}{n}+2;-\frac {a c x^n}{n^2}\right )-\left (n \left (-8 a s+b \left (2 b+\sqrt {(b+1)^2-4 a s}+4\right )+\sqrt {(b+1)^2-4 a s}+2\right )-\sqrt {(b+1)^2-4 a s} \sqrt {n^2 \left ((b+1)^2-4 a s\right )}\right ) \, _0\tilde {F}_1\left (;\frac {n+\sqrt {(b+1)^2-4 a s}}{n};-\frac {a c x^n}{n^2}\right )\right ) \left (x^n\right )^{\frac {\sqrt {(b+1)^2-4 a s}}{n}}+a^{\frac {\sqrt {n^2 \left ((b+1)^2-4 a s\right )}}{n^2}} c^{\frac {\sqrt {n^2 \left ((b+1)^2-4 a s\right )}}{n^2}} n^{\frac {2 \sqrt {(b+1)^2-4 a s}}{n}} c_1 \Gamma \left (-\frac {\sqrt {(b+1)^2-4 a s}}{n}\right ) \left (\left (n \left (8 a s+b \left (-2 b+\sqrt {(b+1)^2-4 a s}-4\right )+\sqrt {(b+1)^2-4 a s}-2\right )+\sqrt {(b+1)^2-4 a s} \sqrt {n^2 \left ((b+1)^2-4 a s\right )}\right ) \, _0\tilde {F}_1\left (;\frac {n-\sqrt {(b+1)^2-4 a s}}{n};-\frac {a c x^n}{n^2}\right )-2 a c \sqrt {(b+1)^2-4 a s} x^n \, _0\tilde {F}_1\left (;2-\frac {\sqrt {(b+1)^2-4 a s}}{n};-\frac {a c x^n}{n^2}\right )\right ) \left (x^n\right )^{\frac {\sqrt {n^2 \left ((b+1)^2-4 a s\right )}}{n^2}}\right )}{2 a n^2 x \left (a^{\frac {\sqrt {(b+1)^2-4 a s}}{n}} c^{\frac {\sqrt {(b+1)^2-4 a s}}{n}} n^{\frac {2 \sqrt {n^2 \left ((b+1)^2-4 a s\right )}}{n^2}} J_{\frac {\sqrt {(b+1)^2-4 a s}}{n}}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^n}}{n}\right ) \Gamma \left (\frac {n+\sqrt {(b+1)^2-4 a s}}{n}\right ) \left (x^n\right )^{\frac {\sqrt {(b+1)^2-4 a s}}{n}}+a^{\frac {\sqrt {n^2 \left ((b+1)^2-4 a s\right )}}{n^2}} c^{\frac {\sqrt {n^2 \left ((b+1)^2-4 a s\right )}}{n^2}} n^{\frac {2 \sqrt {(b+1)^2-4 a s}}{n}} J_{-\frac {\sqrt {(b+1)^2-4 a s}}{n}}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^n}}{n}\right ) c_1 \Gamma \left (\frac {n-\sqrt {(b+1)^2-4 a s}}{n}\right ) \left (x^n\right )^{\frac {\sqrt {n^2 \left ((b+1)^2-4 a s\right )}}{n^2}}\right )} \\ y(x)\to -\frac {\frac {\sqrt {n^2 \left ((b+1)^2-4 a s\right )}+n \left (-2 \sqrt {(b+1)^2-4 a s}+b+1\right )}{a n}-\frac {2 c x^n \, _0\tilde {F}_1\left (;2-\frac {\sqrt {(b+1)^2-4 a s}}{n};-\frac {a c x^n}{n^2}\right )}{n \, _0\tilde {F}_1\left (;1-\frac {\sqrt {(b+1)^2-4 a s}}{n};-\frac {a c x^n}{n^2}\right )}}{2 x} \\ \end{align*}