61.16.5 problem 23

Internal problem ID [12177]
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 : 23
Date solved : Wednesday, March 05, 2025 at 05:15:26 PM
CAS classification : [_Riccati]

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

✗ Maple

ode:=diff(y(x),x) = lambda*arctan(x)^n*y(x)^2-b*lambda*x^m*arctan(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]*ArcTan[x]^n*y[x]^2-b*\[Lambda]*x^m*ArcTan[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") 
cg = symbols("cg") 
m = symbols("m") 
n = symbols("n") 
y = Function("y") 
ode = Eq(b*cg*x**m*y(x)*atan(x)**n - b*m*x**(m - 1) - cg*y(x)**2*atan(x)**n + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out