61.13.12 problem 58

Internal problem ID [12153]
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 : 58
Date solved : Sunday, March 30, 2025 at 11:21:39 PM
CAS classification : [_Riccati]

\begin{align*} y^{\prime }&=y^{2}-\frac {\lambda ^{2}}{2}-\frac {3 \lambda ^{2} \tan \left (\lambda x \right )^{2}}{4}+a \cos \left (\lambda x \right )^{2} \sin \left (\lambda x \right )^{n} \end{align*}

✓ Maple. Time used: 0.054 (sec). Leaf size: 1300

ode:=diff(y(x),x) = y(x)^2-1/2*lambda^2-3/4*lambda^2*tan(lambda*x)^2+a*cos(lambda*x)^2*sin(lambda*x)^n; 
dsolve(ode,y(x), singsol=all);
 

\[ \text {Expression too large to display} \]

✗ Mathematica

ode=D[y[x],x]==y[x]^2-1/2*\[Lambda]^2-3/4*\[Lambda]^2*Tan[\[Lambda]*x]^2+a*Cos[\[Lambda]*x]^2*Sin[\[Lambda]*x]^n; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
lambda_ = symbols("lambda_") 
n = symbols("n") 
y = Function("y") 
ode = Eq(-a*sin(lambda_*x)**n*cos(lambda_*x)**2 + 3*lambda_**2*tan(lambda_*x)**2/4 + lambda_**2/2 - y(x)**2 + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -a*sin(lambda_*x)**n*cos(lambda_*x)**2 + 3*lambda_**2*tan(lambda_*x)**2/4 + lambda_**2/2 - y(x)**2 + Derivative(y(x), x) cannot be solved by the lie group method