89.6.48 problem 49

Internal problem ID [24431]
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. Miscellaneous Exercises at page 45
Problem number : 49
Date solved : Thursday, October 02, 2025 at 10:29:29 PM
CAS classification : [_rational]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 152

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

\[ -\frac {\left (\operatorname {arctanh}\left (\frac {\sqrt {\frac {x^{4} y^{2}}{\left (x^{2}-1\right ) y^{2}-x^{2}}}}{\sqrt {-\frac {x^{2}}{\left (x^{2}-1\right ) y^{2}-x^{2}}}\, x^{2}}\right )-c_1 \right ) \sqrt {\frac {x^{4} y^{2}}{\left (x^{2}-1\right ) y^{2}-x^{2}}}-\sqrt {-\frac {x^{2}}{\left (x^{2}-1\right ) y^{2}-x^{2}}}}{\sqrt {\frac {x^{4} y^{2}}{\left (x^{2}-1\right ) y^{2}-x^{2}}}} = 0 \]

✓ Mathematica. Time used: 0.177 (sec). Leaf size: 30

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

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

✗ Sympy

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

Timed Out