Internal problem ID [10612]
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: 9.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime }-y^{2} f \left (x \right )-g \left (x \right ) y=-f \left (x \right ) a^{2}-a g \left (x \right )} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 45
dsolve(diff(y(x),x)=f(x)*y(x)^2+g(x)*y(x)-a^2*f(x)-a*g(x),y(x), singsol=all)
\[ y \left (x \right ) = a -\frac {{\mathrm e}^{\int g \left (x \right )d x +2 a \left (\int f \left (x \right )d x \right )}}{\int {\mathrm e}^{\int g \left (x \right )d x +2 a \left (\int f \left (x \right )d x \right )} f \left (x \right )d x -c_{1}} \]
✓ Solution by Mathematica
Time used: 1.122 (sec). Leaf size: 201
DSolve[y'[x]==f[x]*y[x]^2+g[x]*y[x]-a^2*f[x]-a*g[x],y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^x-\frac {\exp \left (-\int _1^{K[2]}(-2 a f(K[1])-g(K[1]))dK[1]\right ) (a f(K[2])+y(x) f(K[2])+g(K[2]))}{a-y(x)}dK[2]+\int _1^{y(x)}\left (-\int _1^x\left (-\frac {\exp \left (-\int _1^{K[2]}(-2 a f(K[1])-g(K[1]))dK[1]\right ) f(K[2])}{a-K[3]}-\frac {\exp \left (-\int _1^{K[2]}(-2 a f(K[1])-g(K[1]))dK[1]\right ) (a f(K[2])+K[3] f(K[2])+g(K[2]))}{(a-K[3])^2}\right )dK[2]-\frac {\exp \left (-\int _1^x(-2 a f(K[1])-g(K[1]))dK[1]\right )}{(K[3]-a)^2}\right )dK[3]=c_1,y(x)\right ] \]