Internal problem ID [10376]
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: 36.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, _Riccati]
\[ \boxed {y^{\prime } x -a y^{2}-\left (n +b \,x^{n}\right ) y=c \,x^{2 n}} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 81
dsolve(x*diff(y(x),x)=a*y(x)^2+(n+b*x^n)*y(x)+c*x^(2*n),y(x), singsol=all)
\[ y \left (x \right ) = \frac {x^{2 n -1} \left (\sqrt {4 b^{2} a c -b^{4}}\, \tan \left (\frac {\sqrt {4 b^{2} a c -b^{4}}\, \left (b \,x^{n}+c_{1} n \right )}{2 b^{2} n}\right )-b^{2}\right ) x^{-n +1}}{2 a b} \]
✓ Solution by Mathematica
Time used: 1.07 (sec). Leaf size: 114
DSolve[x*y'[x]==a*y[x]^2+(n+b*x^n)*y[x]+c*x^(2*n),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {x^n \left (-b+\frac {\sqrt {b^2-4 a c} \left (-e^{\frac {x^n \sqrt {b^2-4 a c}}{n}}+c_1\right )}{e^{\frac {x^n \sqrt {b^2-4 a c}}{n}}+c_1}\right )}{2 a} y(x)\to \frac {x^n \left (\sqrt {b^2-4 a c}-b\right )}{2 a} \end{align*}