61.13.10 problem 56

Internal problem ID [12151]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.2. Riccati Equation. subsection 1.2.6-5. Equations containing combinations of trigonometric functions.
Problem number : 56
Date solved : Sunday, March 30, 2025 at 11:20:54 PM
CAS classification : [_Riccati]

\begin{align*} y^{\prime }&=y^{2}-2 \lambda ^{2} \tan \left (x \right )^{2}-2 \lambda ^{2} \cot \left (\lambda x \right )^{2} \end{align*}

✗ Maple

ode:=diff(y(x),x) = y(x)^2-2*lambda^2*tan(x)^2-2*lambda^2*cot(lambda*x)^2; 
dsolve(ode,y(x), singsol=all);
 

\[ \text {No solution found} \]

✗ Mathematica

ode=D[y[x],x]==y[x]^2-2*\[Lambda]^2*Tan[x]^2-2*\[Lambda]^2*Cot[\[Lambda]*x]^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

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

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