55.21.8 problem 8

Internal problem ID [13535]
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 : 8
Date solved : Wednesday, October 01, 2025 at 04:04:54 PM
CAS classification : [_Riccati]

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

✗ Maple

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

\[ \text {No solution found} \]

✗ Mathematica

ode=D[y[x],x]==y[x]^2-\[Lambda]^2/4+Exp[2*\[Lambda]*x]/(c*Exp[\[Lambda]*x]+d)^4*f[(a*Exp[\[Lambda]*x]+b)/(c*Exp[\[Lambda]*x]+d)]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

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

Timed Out