Internal problem ID [9849]
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: 3.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime }-y^{2}-x f \left (x \right ) y-f \left (x \right )=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 49
dsolve(diff(y(x),x)=y(x)^2+x*f(x)*y(x)+f(x),y(x), singsol=all)
\[ y \left (x \right ) = \frac {{\mathrm e}^{\int \frac {f \left (x \right ) x^{2}-2}{x}d x}}{c_{1} -\left (\int {\mathrm e}^{\int \frac {f \left (x \right ) x^{2}-2}{x}d x}d x \right )}-\frac {1}{x} \]
✓ Solution by Mathematica
Time used: 0.508 (sec). Leaf size: 76
DSolve[y'[x]==y[x]^2+x*f[x]*y[x]+f[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {x+\frac {\exp \left (-\int _1^x-f(K[5]) K[5]dK[5]\right )}{\int _1^x\frac {\exp \left (-\int _1^{K[6]}-f(K[5]) K[5]dK[5]\right )}{K[6]^2}dK[6]+c_1}}{x^2} \\ y(x)\to -\frac {1}{x} \\ \end{align*}