Internal problem ID [10340]
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: 10.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime }-a \,x^{n} y^{2}=b m \,x^{m -1}-a \,b^{2} x^{n +2 m}} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 522
dsolve(diff(y(x),x)=a*x^n*y(x)^2+b*m*x^(m-1)-a*b^2*x^(n+2*m),y(x), singsol=all)
\[ y \left (x \right ) = \frac {x^{-n -1} \left (-\frac {3 \left (a b \left (m +\frac {4 n}{3}+\frac {4}{3}\right ) x^{n +1-\frac {m}{2}}+\frac {x^{-\frac {3 m}{2}} \left (m +2 n +2\right ) \left (1+m +n \right )}{3}\right ) c_{1} \left (m +2 n +2\right ) {\mathrm e}^{\frac {a b \,x^{1+m +n}}{1+m +n}} \operatorname {WhittakerM}\left (\frac {m +2 n +2}{2 n +2 m +2}, \frac {2 m +3 n +3}{2 n +2 m +2}, -\frac {2 a b \,x^{1+m +n}}{1+m +n}\right )}{2}+\left (a^{2} b^{2} x^{2 n +2+\frac {m}{2}}-\frac {\left (x^{n +1-\frac {m}{2}} a b +x^{-\frac {3 m}{2}} \left (1+m +n \right )\right ) \left (m +2 n +2\right )}{2}\right ) \left (1+m +n \right ) c_{1} {\mathrm e}^{\frac {a b \,x^{1+m +n}}{1+m +n}} \operatorname {WhittakerM}\left (-\frac {m}{2 n +2 m +2}, \frac {2 m +3 n +3}{2 n +2 m +2}, -\frac {2 a b \,x^{1+m +n}}{1+m +n}\right )+{\mathrm e}^{\frac {2 a b \,x^{1+m +n}}{1+m +n}} \left (m +\frac {3 n}{2}+\frac {3}{2}\right ) c_{1} \left (m +2 n +2\right )^{2} x^{-\frac {3 m}{2}} \left (-\frac {2 a b \,x^{1+m +n}}{1+m +n}\right )^{\frac {3 m +4 n +4}{2 n +2 m +2}}+a b \,x^{m +2 n +2}\right )}{\left (-\frac {{\mathrm e}^{\frac {a b \,x^{1+m +n}}{1+m +n}} x^{-\frac {3 m}{2}} c_{1} \left (m +2 n +2\right )^{2} \operatorname {WhittakerM}\left (\frac {m +2 n +2}{2 n +2 m +2}, \frac {2 m +3 n +3}{2 n +2 m +2}, -\frac {2 a b \,x^{1+m +n}}{1+m +n}\right )}{2}+\left (1+m +n \right ) c_{1} \left (x^{n +1-\frac {m}{2}} a b -\frac {x^{-\frac {3 m}{2}} \left (m +2 n +2\right )}{2}\right ) {\mathrm e}^{\frac {a b \,x^{1+m +n}}{1+m +n}} \operatorname {WhittakerM}\left (-\frac {m}{2 n +2 m +2}, \frac {2 m +3 n +3}{2 n +2 m +2}, -\frac {2 a b \,x^{1+m +n}}{1+m +n}\right )+x^{n +1}\right ) a} \]
✓ Solution by Mathematica
Time used: 2.322 (sec). Leaf size: 306
DSolve[y'[x]==a*x^n*y[x]^2+b*m*x^(m-1)-a*b^2*x^(n+2*m),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {2^{\frac {n+1}{m+n+1}} (m+n+1) \left (-\frac {a b x^{m+n+1}}{m+n+1}\right )^{\frac {n+1}{m+n+1}} \left (a b x^m-c_1 e^{\frac {2 a b x^{m+n+1}}{m+n+1}}\right )-a b c_1 x^{m+n+1} \Gamma \left (\frac {n+1}{m+n+1},-\frac {2 a b x^{m+n+1}}{m+n+1}\right )}{a \left (2^{\frac {n+1}{m+n+1}} (m+n+1) \left (-\frac {a b x^{m+n+1}}{m+n+1}\right )^{\frac {n+1}{m+n+1}}-c_1 x^{n+1} \Gamma \left (\frac {n+1}{m+n+1},-\frac {2 a b x^{m+n+1}}{m+n+1}\right )\right )} \\ y(x)\to b x^m-\frac {b 2^{\frac {n+1}{m+n+1}} x^m e^{\frac {2 a b x^{m+n+1}}{m+n+1}} \left (-\frac {a b x^{m+n+1}}{m+n+1}\right )^{-\frac {m}{m+n+1}}}{\Gamma \left (\frac {n+1}{m+n+1},-\frac {2 a b x^{m+n+1}}{m+n+1}\right )} \\ \end{align*}