61.21.7 problem 7

Internal problem ID [12239]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.2. Riccati Equation. subsection 1.2.9. Some Transformations
Problem number : 7
Date solved : Wednesday, March 05, 2025 at 06:13:19 PM
CAS classification : [_Riccati]

\begin{align*} y^{\prime }&=y^{2}+{\mathrm e}^{2 \lambda x} f \left ({\mathrm e}^{\lambda x}\right )-\frac {\lambda ^{2}}{4} \end{align*}

✗ Maple

ode:=diff(y(x),x) = y(x)^2+exp(2*lambda*x)*f(exp(lambda*x))-1/4*lambda^2; 
dsolve(ode,y(x), singsol=all);
 

\[ \text {No solution found} \]

✗ Mathematica

ode=D[y[x],x]==y[x]^2+Exp[2*\[Lambda]*x]*f[Exp[\[Lambda]*x]]-1/4*\[Lambda]^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

from sympy import * 
x = symbols("x") 
cg = symbols("cg") 
y = Function("y") 
f = Function("f") 
ode = Eq(cg**2/4 - f(exp(cg*x))*exp(2*cg*x) - y(x)**2 + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE cg**2/4 - f(exp(cg*x))*exp(2*cg*x) - y(x)**2 + Derivative(y(x), x) cannot be solved by the lie group method