61.20.5 problem 38

Internal problem ID [12228]
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
Date solved : Friday, March 14, 2025 at 04:37:28 AM
CAS classification : [_Riccati]

\begin{align*} y^{\prime }&=\frac {f^{\prime }\left (x \right ) y^{2}}{g \left (x \right )}-\frac {g^{\prime }\left (x \right )}{f \left (x \right )} \end{align*}

✓ Maple. Time used: 0.008 (sec). Leaf size: 58

ode:=diff(y(x),x) = diff(f(x),x)/g(x)*y(x)^2-diff(g(x),x)/f(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {-f g \left (x \right ) \left (\int \frac {f^{\prime }}{g \left (x \right ) f^{2}}d x \right )-c_{1} f g \left (x \right )-1}{f^{2} \left (\int \frac {f^{\prime }}{g \left (x \right ) f^{2}}d x +c_{1} \right )} \]

✓ Mathematica. Time used: 0.211 (sec). Leaf size: 160

ode=D[y[x],x]==D[ f[x],x]/g[x]*y[x]^2-D[ g[x],x]/f[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\int _1^{y(x)}\left (\frac {1}{(g(x)+f(x) K[2])^2}-\int _1^x\left (\frac {2 \left (f(K[1]) K[2]^2 f''(K[1])-g(K[1]) g''(K[1])\right )}{g(K[1]) (g(K[1])+f(K[1]) K[2])^3}-\frac {2 K[2] f''(K[1])}{g(K[1]) (g(K[1])+f(K[1]) K[2])^2}\right )dK[1]\right )dK[2]+\int _1^x-\frac {f(K[1]) y(x)^2 f''(K[1])-g(K[1]) g''(K[1])}{f(K[1]) g(K[1]) (g(K[1])+f(K[1]) y(x))^2}dK[1]=c_1,y(x)\right ] \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
f = Function("f") 
g = Function("g") 
ode = Eq(Derivative(y(x), x) - y(x)**2*Derivative(f(x), x)/g(x) + Derivative(g(x), x)/f(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - y(x)**2*Derivative(f(x), x)/g(x) + Derivative(g(x), x)/f(x) cannot be solved by the lie group method