64.4.13 problem 13

Internal problem ID [13302]
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 : Tuesday, January 28, 2025 at 05:17:00 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*}

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 42

dsolve((x^3+y(x)^2*sqrt(x^2+y(x)^2))-x*y(x)*sqrt(x^2+y(x)^2)*diff(y(x),x)=0,y(x), singsol=all)
 

\[ \frac {\left (-y^{2}-x^{2}\right ) \sqrt {x^{2}+y^{2}}-x^{3} \left (c_{1} -3 \ln \left (x \right )\right )}{x^{3}} = 0 \]

✓ Solution by Mathematica

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

DSolve[(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,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*}