61.19.16 problem 16

Internal problem ID [12206]
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 : 16
Date solved : Monday, March 31, 2025 at 04:33:27 AM
CAS classification : [_Riccati]

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

✗ Maple

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

\[ \text {No solution found} \]

✗ Mathematica

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

Not solved

✗ Sympy

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

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