61.18.5 problem 33

Internal problem ID [12187]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.2. Riccati Equation. subsection 1.2.7-3. Equations containing arctangent.
Problem number : 33
Date solved : Wednesday, March 05, 2025 at 05:24:54 PM
CAS classification : [_Riccati]

\begin{align*} y^{\prime }&=\lambda \operatorname {arccot}\left (x \right )^{n} y^{2}+\beta m \,x^{m -1}-\lambda \,\beta ^{2} x^{2 m} \operatorname {arccot}\left (x \right )^{n} \end{align*}

✗ Maple

ode:=diff(y(x),x) = lambda*arccot(x)^n*y(x)^2+beta*m*x^(m-1)-lambda*beta^2*x^(2*m)*arccot(x)^n; 
dsolve(ode,y(x), singsol=all);
 

\[ \text {No solution found} \]

✗ Mathematica

ode=D[y[x],x]==\[Lambda]*ArcCot[x]^n*y[x]^2+\[Beta]*m*x^(m-1)-\[Lambda]*\[Beta]^2*x^(2*m)*ArcCot[x]^n; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

from sympy import * 
x = symbols("x") 
BETA = symbols("BETA") 
lambda_ = symbols("lambda_") 
m = symbols("m") 
n = symbols("n") 
y = Function("y") 
ode = Eq(BETA**2*lambda_*x**(2*m)*(-atan(x) + pi/2)**n - BETA*m*x**(m - 1) - lambda_*(-atan(x) + pi/2)**n*y(x)**2 + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out