Internal problem ID [3949]
Book: Ordinary Differential Equations, By Tenenbaum and Pollard. Dover, NY 1963
Section: Chapter 2. Special types of differential equations of the first kind. Lesson 9
Problem number: Exact Differential equations. Exercise 9.6, page 79.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, class A], _exact, _rational, _dAlembert]
Solve \begin {gather*} \boxed {2 x y+\left (x^{2}+y^{2}\right ) y^{\prime }=0} \end {gather*}
✓ Solution by Maple
Time used: 0.048 (sec). Leaf size: 257
dsolve(2*x*y(x)+(x^2+y(x)^2)*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\frac {\left (4+4 \sqrt {4 x^{6} c_{1}^{3}+1}\right )^{\frac {1}{3}}}{2}-\frac {2 x^{2} c_{1}}{\left (4+4 \sqrt {4 x^{6} c_{1}^{3}+1}\right )^{\frac {1}{3}}}}{\sqrt {c_{1}}} \\ y \relax (x ) = \frac {-\frac {\left (4+4 \sqrt {4 x^{6} c_{1}^{3}+1}\right )^{\frac {1}{3}}}{4}+\frac {x^{2} c_{1}}{\left (4+4 \sqrt {4 x^{6} c_{1}^{3}+1}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (4+4 \sqrt {4 x^{6} c_{1}^{3}+1}\right )^{\frac {1}{3}}}{2}+\frac {2 x^{2} c_{1}}{\left (4+4 \sqrt {4 x^{6} c_{1}^{3}+1}\right )^{\frac {1}{3}}}\right )}{2}}{\sqrt {c_{1}}} \\ y \relax (x ) = \frac {-\frac {\left (4+4 \sqrt {4 x^{6} c_{1}^{3}+1}\right )^{\frac {1}{3}}}{4}+\frac {x^{2} c_{1}}{\left (4+4 \sqrt {4 x^{6} c_{1}^{3}+1}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (4+4 \sqrt {4 x^{6} c_{1}^{3}+1}\right )^{\frac {1}{3}}}{2}+\frac {2 x^{2} c_{1}}{\left (4+4 \sqrt {4 x^{6} c_{1}^{3}+1}\right )^{\frac {1}{3}}}\right )}{2}}{\sqrt {c_{1}}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 2.447 (sec). Leaf size: 367
DSolve[2*x*y[x]+(x^2+y[x]^2)*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt [3]{\sqrt {4 x^6+e^{6 c_1}}+e^{3 c_1}}}{\sqrt [3]{2}}-\frac {\sqrt [3]{2} x^2}{\sqrt [3]{\sqrt {4 x^6+e^{6 c_1}}+e^{3 c_1}}} \\ y(x)\to \frac {2 \sqrt [3]{-2} x^2+(-2)^{2/3} \left (\sqrt {4 x^6+e^{6 c_1}}+e^{3 c_1}\right ){}^{2/3}}{2 \sqrt [3]{\sqrt {4 x^6+e^{6 c_1}}+e^{3 c_1}}} \\ y(x)\to -\frac {2 (-1)^{2/3} x^2+\sqrt [3]{-2} \left (\sqrt {4 x^6+e^{6 c_1}}+e^{3 c_1}\right ){}^{2/3}}{2^{2/3} \sqrt [3]{\sqrt {4 x^6+e^{6 c_1}}+e^{3 c_1}}} \\ y(x)\to 0 \\ y(x)\to \frac {\left (-1-i \sqrt {3}\right ) \sqrt [3]{x^6}+\left (1-i \sqrt {3}\right ) x^2}{2 \sqrt [6]{x^6}} \\ y(x)\to \frac {i \left (\sqrt {3}+i\right ) \sqrt [3]{x^6}+\left (1+i \sqrt {3}\right ) x^2}{2 \sqrt [6]{x^6}} \\ y(x)\to \sqrt [6]{x^6}-\frac {\left (x^6\right )^{5/6}}{x^4} \\ \end{align*}