61.13.1 problem 47

Internal problem ID [12142]
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 : 47
Date solved : Sunday, March 30, 2025 at 11:15:42 PM
CAS classification : [_Riccati]

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

✗ Maple

ode:=diff(y(x),x) = y(x)^2+lambda^2+c*sin(lambda*x)^n*cos(lambda*x)^(-n-4); 
dsolve(ode,y(x), singsol=all);
 

\[ \text {No solution found} \]

✗ Mathematica

ode=D[y[x],x]==y[x]^2+\[Lambda]^2+c*Sin[\[Lambda]*x]^n*Cos[\[Lambda]*x]^(-n-4); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

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

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