Internal problem ID [10350]
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: 1166
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)
\[ \text {Expression too large to display} \]
✓ 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*}