Internal problem ID [10641]
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]
\[ \boxed {y^{\prime }-\frac {f^{\prime }\left (x \right ) y^{2}}{g \left (x \right )}=-\frac {g^{\prime }\left (x \right )}{f \left (x \right )}} \]
✓ Solution by Maple
Time used: 0.015 (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)
\[ y \left (x \right ) = -\frac {g \left (x \right ) \left (\int \frac {\frac {d}{d x}f \left (x \right )}{g \left (x \right ) f \left (x \right )^{2}}d x \right ) f \left (x \right )+c_{1} f \left (x \right ) g \left (x \right )+1}{f \left (x \right )^{2} \left (\int \frac {\frac {d}{d x}f \left (x \right )}{g \left (x \right ) f \left (x \right )^{2}}d x +c_{1} \right )} \]
✓ Solution by Mathematica
Time used: 0.347 (sec). Leaf size: 160
DSolve[y'[x]==f'[x]/g[x]*y[x]^2-g'[x]/f[x],y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}\left (\frac {1}{(g(x)+f(x) K[2])^2}-\int _1^x\left (\frac {2 \left (f(K[1]) K[2]^2 f'(K[1])-g(K[1]) g'(K[1])\right )}{g(K[1]) (g(K[1])+f(K[1]) K[2])^3}-\frac {2 K[2] f'(K[1])}{g(K[1]) (g(K[1])+f(K[1]) K[2])^2}\right )dK[1]\right )dK[2]+\int _1^x-\frac {f(K[1]) y(x)^2 f'(K[1])-g(K[1]) g'(K[1])}{f(K[1]) g(K[1]) (g(K[1])+f(K[1]) y(x))^2}dK[1]=c_1,y(x)\right ] \]