Internal problem ID [9859]
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]
Solve \begin {gather*} \boxed {y^{\prime }-y^{2} f \relax (x )-g \relax (x ) y+a^{2} f \relax (x )+a g \relax (x )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (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)
\[ y \relax (x ) = a -\frac {{\mathrm e}^{\int g \relax (x )d x +2 a \left (\int f \relax (x )d x \right )}}{\int {\mathrm e}^{\int g \relax (x )d x +2 a \left (\int f \relax (x )d x \right )} f \relax (x )d x -c_{1}} \]
✓ Solution by Mathematica
Time used: 0.623 (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]
\[ \text {Solve}\left [\int _1^x-\frac {\exp \left (-\int _1^{K[6]}(-2 a f(K[5])-g(K[5]))dK[5]\right ) (a f(K[6])+y(x) f(K[6])+g(K[6]))}{a-y(x)}dK[6]+\int _1^{y(x)}\left (-\int _1^x\left (-\frac {\exp \left (-\int _1^{K[6]}(-2 a f(K[5])-g(K[5]))dK[5]\right ) f(K[6])}{a-K[7]}-\frac {\exp \left (-\int _1^{K[6]}(-2 a f(K[5])-g(K[5]))dK[5]\right ) (a f(K[6])+K[7] f(K[6])+g(K[6]))}{(a-K[7])^2}\right )dK[6]-\frac {\exp \left (-\int _1^x(-2 a f(K[5])-g(K[5]))dK[5]\right )}{(K[7]-a)^2}\right )dK[7]=c_1,y(x)\right ] \]