61.19.18 problem 18

Internal problem ID [12208]
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 : 18
Date solved : Monday, March 31, 2025 at 04:33:39 AM
CAS classification : [_Riccati]

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

✗ Maple

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

\[ \text {No solution found} \]

✗ Mathematica

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

Not solved

✗ Sympy

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

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