Internal problem ID [9595]
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: 8.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
Solve \begin {gather*} \boxed {y^{\prime }-a \,x^{n} y^{2}-b \,x^{m}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.002 (sec). Leaf size: 177
dsolve(diff(y(x),x)=a*x^n*y(x)^2+b*x^m,y(x), singsol=all)
\[ y \relax (x ) = \frac {\left (\BesselY \left (\frac {m +1}{m +n +2}, \frac {2 \sqrt {a b}\, x^{\frac {m}{2}+\frac {n}{2}+1}}{m +n +2}\right ) c_{1}+\BesselJ \left (\frac {m +1}{m +n +2}, \frac {2 \sqrt {a b}\, x^{\frac {m}{2}+\frac {n}{2}+1}}{m +n +2}\right )\right ) x^{\frac {m}{2}+\frac {n}{2}+1} \sqrt {a b}\, x^{-n}}{\left (\BesselY \left (-\frac {n +1}{m +n +2}, \frac {2 \sqrt {a b}\, x^{\frac {m}{2}+\frac {n}{2}+1}}{m +n +2}\right ) c_{1}+\BesselJ \left (-\frac {n +1}{m +n +2}, \frac {2 \sqrt {a b}\, x^{\frac {m}{2}+\frac {n}{2}+1}}{m +n +2}\right )\right ) a x} \]
✓ Solution by Mathematica
Time used: 2.347 (sec). Leaf size: 827
DSolve[y'[x]==a*x^n*y[x]^2+b*x^m,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {(n+1) x^{-n-1} \left ((m+n+1)^{\frac {2 (n+1)}{m+n+2}} \Gamma \left (\frac {n+1}{m+n+2}\right ) \left (-\sqrt {a} \sqrt {b} (m+n+1) \left (x^{m+n+1}\right )^{\frac {m+n+2}{2 (m+n+1)}} J_{-\frac {m+1}{m+n+2}}\left (\frac {2 \sqrt {a} \sqrt {b} (m+n+1) \left (x^{m+n+1}\right )^{\frac {1}{2} \left (1+\frac {1}{m+n+1}\right )}}{\sqrt {(m+n+1)^2} (m+n+2)}\right )+\sqrt {a} \sqrt {b} (m+n+1) \left (x^{m+n+1}\right )^{\frac {m+n+2}{2 (m+n+1)}} J_{\frac {m+2 n+3}{m+n+2}}\left (\frac {2 \sqrt {a} \sqrt {b} (m+n+1) \left (x^{m+n+1}\right )^{\frac {1}{2} \left (1+\frac {1}{m+n+1}\right )}}{\sqrt {(m+n+1)^2} (m+n+2)}\right )-(n+1) \sqrt {(m+n+1)^2} J_{\frac {n+1}{m+n+2}}\left (\frac {2 \sqrt {a} \sqrt {b} (m+n+1) \left (x^{m+n+1}\right )^{\frac {1}{2} \left (1+\frac {1}{m+n+1}\right )}}{\sqrt {(m+n+1)^2} (m+n+2)}\right )\right )-2 \sqrt {a} \sqrt {b} c_1 (m+n+1) \left ((m+n+1)^2\right )^{\frac {n+1}{m+n+2}} \left (x^{m+n+1}\right )^{\frac {1}{2} \left (\frac {1}{m+n+1}+1\right )} \Gamma \left (-\frac {n+1}{m+n+2}\right ) J_{\frac {m+1}{m+n+2}}\left (\frac {2 \sqrt {a} \sqrt {b} (m+n+1) \left (x^{m+n+1}\right )^{\frac {1}{2} \left (1+\frac {1}{m+n+1}\right )}}{\sqrt {(m+n+1)^2} (m+n+2)}\right )\right )}{2 a \sqrt {(m+n+1)^2} \left ((n+1) (m+n+1)^{\frac {2 (n+1)}{m+n+2}} \Gamma \left (\frac {n+1}{m+n+2}\right ) J_{\frac {n+1}{m+n+2}}\left (\frac {2 \sqrt {a} \sqrt {b} (m+n+1) \left (x^{m+n+1}\right )^{\frac {1}{2} \left (1+\frac {1}{m+n+1}\right )}}{\sqrt {(m+n+1)^2} (m+n+2)}\right )+c_1 (m+n+2) \left ((m+n+1)^2\right )^{\frac {n+1}{m+n+2}} \Gamma \left (\frac {m+1}{m+n+2}\right ) J_{-\frac {n+1}{m+n+2}}\left (\frac {2 \sqrt {a} \sqrt {b} (m+n+1) \left (x^{m+n+1}\right )^{\frac {1}{2} \left (1+\frac {1}{m+n+1}\right )}}{\sqrt {(m+n+1)^2} (m+n+2)}\right )\right )} \\ y(x)\to \frac {x^{-n-1} \left (\frac {\frac {a b \left (x^{m+n+1}\right )^{\frac {1}{m+n+1}+1} \, _0F_1\left (;\frac {m+1}{m+n+2}+1;-\frac {a b \left (x^{m+n+1}\right )^{1+\frac {1}{m+n+1}}}{(m+n+2)^2}\right )}{m+1}+(n+1) \, _0F_1\left (;-\frac {n+1}{m+n+2};-\frac {a b \left (x^{m+n+1}\right )^{1+\frac {1}{m+n+1}}}{(m+n+2)^2}\right )}{\, _0F_1\left (;\frac {m+1}{m+n+2};-\frac {a b \left (x^{m+n+1}\right )^{1+\frac {1}{m+n+1}}}{(m+n+2)^2}\right )}-n-1\right )}{2 a} \\ \end{align*}