Internal problem ID [10604]
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: 1.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime }-y^{2}-f \left (x \right ) y=-a^{2}-a f \left (x \right )} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 36
dsolve(diff(y(x),x)=y(x)^2+f(x)*y(x)-a^2-a*f(x),y(x), singsol=all)
\[ y \left (x \right ) = a -\frac {{\mathrm e}^{\int f \left (x \right )d x +2 a x}}{\int {\mathrm e}^{\int f \left (x \right )d x +2 a x}d x -c_{1}} \]
✓ Solution by Mathematica
Time used: 0.719 (sec). Leaf size: 166
DSolve[y'[x]==y[x]^2+f[x]*y[x]-a^2-a*f[x],y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^x\frac {\exp \left (-\int _1^{K[2]}(-2 a-f(K[1]))dK[1]\right ) (a+f(K[2])+y(x))}{a-y(x)}dK[2]+\int _1^{y(x)}\left (\frac {\exp \left (-\int _1^x(-2 a-f(K[1]))dK[1]\right )}{(K[3]-a)^2}-\int _1^x\left (\frac {\exp \left (-\int _1^{K[2]}(-2 a-f(K[1]))dK[1]\right ) (a+f(K[2])+K[3])}{(a-K[3])^2}+\frac {\exp \left (-\int _1^{K[2]}(-2 a-f(K[1]))dK[1]\right )}{a-K[3]}\right )dK[2]\right )dK[3]=c_1,y(x)\right ] \]