Internal problem ID [9888]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second
edition
Section: Chapter 1, section 1.2. Riccati Equation. subsection 1.2.8-2. Equations containing arbitrary
functions and their derivatives.
Problem number: 38.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {f^{\prime }\relax (x ) y^{2}}{g \relax (x )}+\frac {g^{\prime }\relax (x )}{f \relax (x )}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 57
dsolve(diff(y(x),x)=diff(f(x),x)/g(x)*y(x)^2-diff(g(x),x)/f(x),y(x), singsol=all)
\[ y \relax (x ) = -\frac {f \relax (x ) \left (\int \frac {\frac {d}{d x}f \relax (x )}{f \relax (x )^{2} g \relax (x )}d x \right ) g \relax (x )+f \relax (x ) g \relax (x ) c_{1}+1}{f \relax (x )^{2} \left (\int \frac {\frac {d}{d x}f \relax (x )}{f \relax (x )^{2} g \relax (x )}d x +c_{1}\right )} \]
✓ Solution by Mathematica
Time used: 0.29 (sec). Leaf size: 160
DSolve[y'[x]==f'[x]/g[x]*y[x]^2-g'[x]/f[x],y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}\left (\frac {1}{(g(x)+f(x) K[2])^2}-\int _1^x\left (\frac {2 \left (f(K[1]) K[2]^2 f'(K[1])-g(K[1]) g'(K[1])\right )}{g(K[1]) (g(K[1])+f(K[1]) K[2])^3}-\frac {2 K[2] f'(K[1])}{g(K[1]) (g(K[1])+f(K[1]) K[2])^2}\right )dK[1]\right )dK[2]+\int _1^x-\frac {f(K[1]) y(x)^2 f'(K[1])-g(K[1]) g'(K[1])}{f(K[1]) g(K[1]) (g(K[1])+f(K[1]) y(x))^2}dK[1]=c_1,y(x)\right ] \]