26.4.17 problem Recognizable Exact Differential equations. Integrating factors. Exercise 10.9, page 90

Internal problem ID [6963]
Book : Ordinary Differential Equations, By Tenenbaum and Pollard. Dover, NY 1963
Section : Chapter 2. Special types of differential equations of the first kind. Lesson 10
Problem number : Recognizable Exact Differential equations. Integrating factors. Exercise 10.9, page 90
Date solved : Tuesday, September 30, 2025 at 04:07:20 PM
CAS classification : [_exact]

\begin{align*} \arctan \left (x y\right )+\frac {x y-2 x y^{2}}{1+x^{2} y^{2}}+\frac {\left (x^{2}-2 x^{2} y\right ) y^{\prime }}{1+x^{2} y^{2}}&=0 \end{align*}

✓ Maple. Time used: 0.044 (sec). Leaf size: 22

ode:=arctan(x*y(x))+(x*y(x)-2*x*y(x)^2)/(1+x^2*y(x)^2)+(x^2-2*x^2*y(x))/(1+x^2*y(x)^2)*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\tan \left (\operatorname {RootOf}\left (\textit {\_Z} x -\ln \left (\sec \left (\textit {\_Z} \right )^{2}\right )+c_1 \right )\right )}{x} \]

✓ Mathematica. Time used: 0.131 (sec). Leaf size: 26

ode=(ArcTan[x*y[x]]+(x*y[x]-2*x*y[x]^2)/(1+x^2*y[x]^2))+((x^2-2*x^2*y[x])/(1+x^2*y[x]^2))*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\log \left (x^2 y(x)^2+1\right )-x \arctan (x y(x))=c_1,y(x)\right ] \]

✗ Sympy

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

Timed Out