55.20.2 problem 35

Internal problem ID [13520]
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 : 35
Date solved : Wednesday, October 01, 2025 at 03:56:24 PM
CAS classification : [_Riccati]

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

✗ Maple

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

\[ \text {No solution found} \]

✗ Mathematica

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

Not solved

✗ Sympy

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

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