64.4.13 problem 13

Internal problem ID [13231]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 2, section 2.2 (Separable equations). Exercises page 47
Problem number : 13
Date solved : Monday, March 31, 2025 at 07:42:03 AM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

\begin{align*} x^{3}+y^{2} \sqrt {x^{2}+y^{2}}-x y \sqrt {x^{2}+y^{2}}\, y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 42

ode:=x^3+y(x)^2*(x^2+y(x)^2)^(1/2)-x*y(x)*(x^2+y(x)^2)^(1/2)*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ \frac {\left (-x^{2}-y^{2}\right ) \sqrt {x^{2}+y^{2}}-x^{3} \left (c_1 -3 \ln \left (x \right )\right )}{x^{3}} = 0 \]

✓ Mathematica. Time used: 20.288 (sec). Leaf size: 265

ode=(x^3+y[x]^2*Sqrt[x^2+y[x]^2])-x*y[x]*Sqrt[x^2+y[x]^2]*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to -\sqrt {-x^2-\frac {1}{2} \sqrt [6]{3} \left (\sqrt {3}+3 i\right ) \sqrt [3]{x^6 (\log (x)+c_1){}^2}} \\ y(x)\to \sqrt {-x^2-\frac {1}{2} \sqrt [6]{3} \left (\sqrt {3}+3 i\right ) \sqrt [3]{x^6 (\log (x)+c_1){}^2}} \\ y(x)\to -\sqrt {-x^2-\frac {1}{2} \sqrt [6]{3} \left (\sqrt {3}-3 i\right ) \sqrt [3]{x^6 (\log (x)+c_1){}^2}} \\ y(x)\to \sqrt {-x^2-\frac {1}{2} \sqrt [6]{3} \left (\sqrt {3}-3 i\right ) \sqrt [3]{x^6 (\log (x)+c_1){}^2}} \\ y(x)\to -\sqrt {-x^2+3^{2/3} \sqrt [3]{x^6 (\log (x)+c_1){}^2}} \\ y(x)\to \sqrt {-x^2+3^{2/3} \sqrt [3]{x^6 (\log (x)+c_1){}^2}} \\ \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**3 - x*sqrt(x**2 + y(x)**2)*y(x)*Derivative(y(x), x) + sqrt(x**2 + y(x)**2)*y(x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

TypeError : cannot determine truth value of Relational