Internal problem ID [3968]
Book: Ordinary Differential Equations, By Tenenbaum and Pollard. Dover, NY 1963
Section: Chapter 2. Special types of differential equations of the first kind. Lesson 10
Problem number: Recognizable Exact Differential equations. Integrating factors. Exercise 10.1,
page 90.
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+x^{2}+\left (x^{2}+y^{2}\right ) y^{\prime }=0} \end {gather*}
✓ Solution by Maple
Time used: 0.052 (sec). Leaf size: 417
dsolve((2*x*y(x)+x^2)+(x^2+y(x)^2)*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\frac {\left (4-4 x^{3} c_{1}^{\frac {3}{2}}+4 \sqrt {5 x^{6} c_{1}^{3}-2 x^{3} c_{1}^{\frac {3}{2}}+1}\right )^{\frac {1}{3}}}{2}-\frac {2 x^{2} c_{1}}{\left (4-4 x^{3} c_{1}^{\frac {3}{2}}+4 \sqrt {5 x^{6} c_{1}^{3}-2 x^{3} c_{1}^{\frac {3}{2}}+1}\right )^{\frac {1}{3}}}}{\sqrt {c_{1}}} \\ y \relax (x ) = \frac {-\frac {\left (4-4 x^{3} c_{1}^{\frac {3}{2}}+4 \sqrt {5 x^{6} c_{1}^{3}-2 x^{3} c_{1}^{\frac {3}{2}}+1}\right )^{\frac {1}{3}}}{4}+\frac {x^{2} c_{1}}{\left (4-4 x^{3} c_{1}^{\frac {3}{2}}+4 \sqrt {5 x^{6} c_{1}^{3}-2 x^{3} c_{1}^{\frac {3}{2}}+1}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (4-4 x^{3} c_{1}^{\frac {3}{2}}+4 \sqrt {5 x^{6} c_{1}^{3}-2 x^{3} c_{1}^{\frac {3}{2}}+1}\right )^{\frac {1}{3}}}{2}+\frac {2 x^{2} c_{1}}{\left (4-4 x^{3} c_{1}^{\frac {3}{2}}+4 \sqrt {5 x^{6} c_{1}^{3}-2 x^{3} c_{1}^{\frac {3}{2}}+1}\right )^{\frac {1}{3}}}\right )}{2}}{\sqrt {c_{1}}} \\ y \relax (x ) = \frac {-\frac {\left (4-4 x^{3} c_{1}^{\frac {3}{2}}+4 \sqrt {5 x^{6} c_{1}^{3}-2 x^{3} c_{1}^{\frac {3}{2}}+1}\right )^{\frac {1}{3}}}{4}+\frac {x^{2} c_{1}}{\left (4-4 x^{3} c_{1}^{\frac {3}{2}}+4 \sqrt {5 x^{6} c_{1}^{3}-2 x^{3} c_{1}^{\frac {3}{2}}+1}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (4-4 x^{3} c_{1}^{\frac {3}{2}}+4 \sqrt {5 x^{6} c_{1}^{3}-2 x^{3} c_{1}^{\frac {3}{2}}+1}\right )^{\frac {1}{3}}}{2}+\frac {2 x^{2} c_{1}}{\left (4-4 x^{3} c_{1}^{\frac {3}{2}}+4 \sqrt {5 x^{6} c_{1}^{3}-2 x^{3} c_{1}^{\frac {3}{2}}+1}\right )^{\frac {1}{3}}}\right )}{2}}{\sqrt {c_{1}}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 1.72 (sec). Leaf size: 544
DSolve[(2*x*y[x]+x^2)+(x^2+y[x]^2)*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt [3]{-x^3+\sqrt {5 x^6-2 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}}{\sqrt [3]{2}}-\frac {\sqrt [3]{2} x^2}{\sqrt [3]{-x^3+\sqrt {5 x^6-2 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}} \\ y(x)\to \frac {2 \sqrt [3]{-2} x^2+(-2)^{2/3} \left (-x^3+\sqrt {5 x^6-2 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}\right ){}^{2/3}}{2 \sqrt [3]{-x^3+\sqrt {5 x^6-2 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}} \\ y(x)\to -\frac {2 (-1)^{2/3} x^2+\sqrt [3]{-2} \left (-x^3+\sqrt {5 x^6-2 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}\right ){}^{2/3}}{2^{2/3} \sqrt [3]{-x^3+\sqrt {5 x^6-2 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}} \\ y(x)\to \frac {\left (2 \sqrt {5} \sqrt {x^6}-2 x^3\right )^{2/3}-2 \sqrt [3]{2} x^2}{2 \sqrt [3]{\sqrt {5} \sqrt {x^6}-x^3}} \\ y(x)\to -\frac {2 (-1)^{2/3} x^2+\sqrt [3]{-2} \left (\sqrt {5} \sqrt {x^6}-x^3\right )^{2/3}}{2^{2/3} \sqrt [3]{\sqrt {5} \sqrt {x^6}-x^3}} \\ y(x)\to \frac {2 \sqrt [3]{-2} x^2+(-2)^{2/3} \left (\sqrt {5} \sqrt {x^6}-x^3\right )^{2/3}}{2 \sqrt [3]{\sqrt {5} \sqrt {x^6}-x^3}} \\ \end{align*}