55.18.4 problem 32

Internal problem ID [13483]
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 : 32
Date solved : Wednesday, October 01, 2025 at 03:15:22 PM
CAS classification : [_Riccati]

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

✗ Maple

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

\[ \text {No solution found} \]

✗ Mathematica

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

Not solved

✗ Sympy

from sympy import * 
x = symbols("x") 
b = symbols("b") 
lambda_ = symbols("lambda_") 
m = symbols("m") 
n = symbols("n") 
y = Function("y") 
ode = Eq(b*lambda_*x**m*(-atan(x) + pi/2)**n*y(x) - b*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