61.18.8 problem 36

Internal problem ID [12190]
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 : 36
Date solved : Wednesday, March 05, 2025 at 05:28:45 PM
CAS classification : [_Riccati]

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

✗ Maple

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

\[ \text {No solution found} \]

✗ Mathematica

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

Not solved

✗ Sympy

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

Timed Out