Internal problem ID [9852]
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: 2.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
Solve \begin {gather*} \boxed {y^{\prime }-y^{2} f \relax (x )+a y+b a +b^{2} f \relax (x )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 83
dsolve(diff(y(x),x)=f(x)*y(x)^2-a*y(x)-a*b-b^2*f(x),y(x), singsol=all)
\[ y \relax (x ) = -\frac {\left (\left (\int f \relax (x ) {\mathrm e}^{\int \left (-2 f \relax (x ) b -a \right )d x}d x \right ) {\mathrm e}^{\int \left (2 f \relax (x ) b +a \right )d x} b +c_{1} {\mathrm e}^{\int \left (2 f \relax (x ) b +a \right )d x} b +1\right ) {\mathrm e}^{\int \left (-2 f \relax (x ) b -a \right )d x}}{c_{1}+\int f \relax (x ) {\mathrm e}^{\int \left (-2 f \relax (x ) b -a \right )d x}d x} \]
✓ Solution by Mathematica
Time used: 0.568 (sec). Leaf size: 185
DSolve[y'[x]==f[x]*y[x]^2-a*y[x]-a*b-b^2*f[x],y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^x\frac {\exp \left (-\int _1^{K[2]}(a+2 b f(K[1]))dK[1]\right ) (a+b f(K[2])-f(K[2]) y(x))}{a (b+y(x))}dK[2]+\int _1^{y(x)}\left (\frac {\exp \left (-\int _1^x(a+2 b f(K[1]))dK[1]\right )}{a (b+K[3])^2}-\int _1^x\left (-\frac {\exp \left (-\int _1^{K[2]}(a+2 b f(K[1]))dK[1]\right ) f(K[2])}{a (b+K[3])}-\frac {\exp \left (-\int _1^{K[2]}(a+2 b f(K[1]))dK[1]\right ) (a+b f(K[2])-f(K[2]) K[3])}{a (b+K[3])^2}\right )dK[2]\right )dK[3]=c_1,y(x)\right ] \]