55.19.20 problem 20

Internal problem ID [13505]
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 : 20
Date solved : Wednesday, October 01, 2025 at 03:39:39 PM
CAS classification : [_Riccati]

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

✗ Maple

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

\[ \text {No solution found} \]

✗ Mathematica

ode=D[y[x],x]==f[x]*y[x]^2+2*a*\[Lambda]*x*Exp[\[Lambda]*x^2]-a^2*f[x]*Exp[2*\[Lambda]*x^2]; 
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") 
ode = Eq(a**2*f(x)*exp(2*lambda_*x**2) - 2*a*lambda_*x*exp(lambda_*x**2) - f(x)*y(x)**2 + 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**2) - 2*a*lambda_*x*exp(lambda_*x**2) - f(x)*y(x)**2 + Derivative(y(x), x) cannot be solved by the lie group method