61.19.3 problem 3

Internal problem ID [12272]
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
Date solved : Tuesday, January 28, 2025 at 02:02:44 AM
CAS classification : [_Riccati]

\begin{align*} y^{\prime }&=y^{2}+x f \left (x \right ) y+f \left (x \right ) \end{align*}

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 69

dsolve(diff(y(x),x)=y(x)^2+x*f(x)*y(x)+f(x),y(x), singsol=all)
 

\[ y = \frac {{\mathrm e}^{\int \frac {-2+f x^{2}}{x}d x} x +\int {\mathrm e}^{\int \frac {-2+f x^{2}}{x}d x}d x -c_{1}}{\left (c_{1} -\int {\mathrm e}^{\int \frac {-2+f x^{2}}{x}d x}d x \right ) x} \]

✓ Solution by Mathematica

Time used: 0.709 (sec). Leaf size: 111

DSolve[D[y[x],x]==y[x]^2+x*f[x]*y[x]+f[x],y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\frac {\exp \left (-\int _1^x-f(K[1]) K[1]dK[1]\right )+x \int _1^x\frac {\exp \left (-\int _1^{K[2]}-f(K[1]) K[1]dK[1]\right )}{K[2]^2}dK[2]+c_1 x}{x^2 \left (\int _1^x\frac {\exp \left (-\int _1^{K[2]}-f(K[1]) K[1]dK[1]\right )}{K[2]^2}dK[2]+c_1\right )} \\ y(x)\to -\frac {1}{x} \\ \end{align*}