89.4.10 problem 11

Internal problem ID [24332]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 2. Equations of the first order and first degree. Exercises at page 39
Problem number : 11
Date solved : Thursday, October 02, 2025 at 10:18:28 PM
CAS classification : [_rational]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 121

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

\[ \frac {\operatorname {arctanh}\left (\frac {\sqrt {-\frac {1}{x^{2} \left (x^{2}+y^{2}+1\right )}}\, x^{2}}{\sqrt {\frac {y^{2}}{x^{2}+y^{2}+1}}}\right ) \sqrt {-\frac {1}{x^{2} \left (x^{2}+y^{2}+1\right )}}+c_1 \sqrt {-\frac {1}{x^{2} \left (x^{2}+y^{2}+1\right )}}+\sqrt {\frac {y^{2}}{x^{2}+y^{2}+1}}}{\sqrt {-\frac {1}{x^{2} \left (x^{2}+y^{2}+1\right )}}} = 0 \]

✓ Mathematica. Time used: 0.054 (sec). Leaf size: 20

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

\[ \text {Solve}\left [x y(x)-\arctan \left (\frac {x}{y(x)}\right )=c_1,y(x)\right ] \]

✗ Sympy

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

Timed Out