Internal problem ID [9665]
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: 78.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, class D], _rational, _Riccati]
Solve \begin {gather*} \boxed {\left (a \,x^{n}+b \,x^{m}+c \right ) \left (y^{\prime } x -y\right )+s \,x^{k} \left (y^{2}-\lambda \,x^{2}\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 42
dsolve((a*x^n+b*x^m+c)*(x*diff(y(x),x)-y(x))+s*x^k*(y(x)^2-lambda*x^2)=0,y(x), singsol=all)
\[ y \relax (x ) = \tanh \left (\left (\int \frac {x^{k}}{a \,x^{n}+b \,x^{m}+c}d x \right ) s \sqrt {\lambda }+c_{1} s \sqrt {\lambda }\right ) x \sqrt {\lambda } \]
✓ Solution by Mathematica
Time used: 1.371 (sec). Leaf size: 53
DSolve[(a*x^n+b*x^m+c)*(x*y'[x]-y[x])+s*x^k*(y[x]^2-\[Lambda]*x^2)==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \sqrt {\lambda } (-x) \tanh \left (\sqrt {\lambda } \left (\int _1^x-\frac {s K[1]^k}{b K[1]^m+a K[1]^n+c}dK[1]+c_1\right )\right ) \\ \end{align*}