Internal problem ID [9851]
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]
Solve \begin {gather*} \boxed {y^{\prime }-y^{2}-f \relax (x ) y+a^{2}+a f \relax (x )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.002 (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)
\[ y \relax (x ) = a -\frac {{\mathrm e}^{\int f \relax (x )d x +2 x a}}{\int {\mathrm e}^{\int f \relax (x )d x +2 x a}d x -c_{1}} \]
✓ Solution by Mathematica
Time used: 0.432 (sec). Leaf size: 166
DSolve[y'[x]==y[x]^2+f[x]*y[x]-a^2-a*f[x],y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^x\frac {\exp \left (-\int _1^{K[6]}(-2 a-f(K[5]))dK[5]\right ) (a+f(K[6])+y(x))}{a-y(x)}dK[6]+\int _1^{y(x)}\left (\frac {\exp \left (-\int _1^x(-2 a-f(K[5]))dK[5]\right )}{(K[7]-a)^2}-\int _1^x\left (\frac {\exp \left (-\int _1^{K[6]}(-2 a-f(K[5]))dK[5]\right ) (a+f(K[6])+K[7])}{(a-K[7])^2}+\frac {\exp \left (-\int _1^{K[6]}(-2 a-f(K[5]))dK[5]\right )}{a-K[7]}\right )dK[6]\right )dK[7]=c_1,y(x)\right ] \]