Internal problem ID [9857]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second
edition
Section: Chapter 1, section 1.2. Riccati Equation. subsection 1.2.8-1. Equations containing arbitrary
functions (but not containing their derivatives).
Problem number: 7.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
Solve \begin {gather*} \boxed {x y^{\prime }-f \relax (x ) y^{2}-y n -f \relax (x ) x^{2 n} a=0} \end {gather*}
✓ Solution by Maple
Time used: 0.018 (sec). Leaf size: 35
dsolve(x*diff(y(x),x)=f(x)*y(x)^2+n*y(x)+a*x^(2*n)*f(x),y(x), singsol=all)
\[ y \relax (x ) = -\tan \left (-\sqrt {a}\, \left (\int \frac {f \relax (x ) x^{n}}{x}d x \right )+c_{1}\right ) \sqrt {a}\, x^{n} \]
✓ Solution by Mathematica
Time used: 0.359 (sec). Leaf size: 41
DSolve[x*y'[x]==f[x]*y[x]^2+n*y[x]+a*x^(2*n)*f[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \sqrt {a} x^n \tan \left (\sqrt {a} \int _1^xf(K[1]) K[1]^{n-1}dK[1]+c_1\right ) \\ \end{align*}