61.21.12 problem 12

Internal problem ID [12244]
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 : 12
Date solved : Wednesday, March 05, 2025 at 06:15:46 PM
CAS classification : [_Riccati]

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

✗ Maple

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

\[ \text {No solution found} \]

✗ Mathematica

ode=D[y[x],x]==y[x]^2+\[Lambda]^2+Sin[\[Lambda]*x]^(-4)*f[Cot[\[Lambda]*x]]; 
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 - f(1/tan(cg*x))/sin(cg*x)**4 - y(x)**2 + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -cg**2 - f(1/tan(cg*x))/sin(cg*x)**4 - y(x)**2 + Derivative(y(x), x) cannot be solved by the lie group method