Internal problem ID [3975]
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 10.8,
page 90.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, class G], _rational]
Solve \begin {gather*} \boxed {y \left (2 x +y^{3}\right )-x \left (2 x -y^{3}\right ) y^{\prime }=0} \end {gather*}
✓ Solution by Maple
Time used: 0.09 (sec). Leaf size: 420
dsolve((y(x)*(2*x+y(x)^3))-(x*(2*x-y(x)^3))*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\left (-108 x^{4}+12 \sqrt {81 x^{4}-12 c_{1}^{3}}\, x^{2}+8 c_{1}^{3}\right )^{\frac {1}{3}}}{6 x}+\frac {2 c_{1}^{2}}{3 x \left (-108 x^{4}+12 \sqrt {81 x^{4}-12 c_{1}^{3}}\, x^{2}+8 c_{1}^{3}\right )^{\frac {1}{3}}}+\frac {c_{1}}{3 x} \\ y \relax (x ) = -\frac {\left (-108 x^{4}+12 \sqrt {81 x^{4}-12 c_{1}^{3}}\, x^{2}+8 c_{1}^{3}\right )^{\frac {1}{3}}}{12 x}-\frac {c_{1}^{2}}{3 x \left (-108 x^{4}+12 \sqrt {81 x^{4}-12 c_{1}^{3}}\, x^{2}+8 c_{1}^{3}\right )^{\frac {1}{3}}}+\frac {c_{1}}{3 x}-\frac {i \sqrt {3}\, \left (\frac {\left (-108 x^{4}+12 \sqrt {81 x^{4}-12 c_{1}^{3}}\, x^{2}+8 c_{1}^{3}\right )^{\frac {1}{3}}}{6 x}-\frac {2 c_{1}^{2}}{3 x \left (-108 x^{4}+12 \sqrt {81 x^{4}-12 c_{1}^{3}}\, x^{2}+8 c_{1}^{3}\right )^{\frac {1}{3}}}\right )}{2} \\ y \relax (x ) = -\frac {\left (-108 x^{4}+12 \sqrt {81 x^{4}-12 c_{1}^{3}}\, x^{2}+8 c_{1}^{3}\right )^{\frac {1}{3}}}{12 x}-\frac {c_{1}^{2}}{3 x \left (-108 x^{4}+12 \sqrt {81 x^{4}-12 c_{1}^{3}}\, x^{2}+8 c_{1}^{3}\right )^{\frac {1}{3}}}+\frac {c_{1}}{3 x}+\frac {i \sqrt {3}\, \left (\frac {\left (-108 x^{4}+12 \sqrt {81 x^{4}-12 c_{1}^{3}}\, x^{2}+8 c_{1}^{3}\right )^{\frac {1}{3}}}{6 x}-\frac {2 c_{1}^{2}}{3 x \left (-108 x^{4}+12 \sqrt {81 x^{4}-12 c_{1}^{3}}\, x^{2}+8 c_{1}^{3}\right )^{\frac {1}{3}}}\right )}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 5.236 (sec). Leaf size: 331
DSolve[(y[x]*(2*x+y[x]^3))-(x*(2*x-y[x]^3))*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {2 c_1 \left (1+\frac {c_1}{\sqrt [3]{\frac {27 x^4}{2}+\frac {3}{2} \sqrt {81 x^8+12 c_1{}^3 x^4}+c_1{}^3}}\right )+2^{2/3} \sqrt [3]{27 x^4+3 \sqrt {81 x^8+12 c_1{}^3 x^4}+2 c_1{}^3}}{6 x} \\ y(x)\to -\frac {-\frac {2 \sqrt [3]{-2} c_1{}^2}{\sqrt [3]{27 x^4+3 \sqrt {81 x^8+12 c_1{}^3 x^4}+2 c_1{}^3}}+(-2)^{2/3} \sqrt [3]{27 x^4+3 \sqrt {81 x^8+12 c_1{}^3 x^4}+2 c_1{}^3}+2 c_1}{6 x} \\ y(x)\to \frac {\sqrt [3]{27 x^4+3 \sqrt {81 x^8+12 c_1{}^3 x^4}+2 c_1{}^3} \text {Root}\left [\text {$\#$1}^3+32\&,3\right ]+c_1 \left (-4+\frac {c_1 \text {Root}\left [\text {$\#$1}^3+128\&,2\right ]}{\sqrt [3]{27 x^4+3 \sqrt {81 x^8+12 c_1{}^3 x^4}+2 c_1{}^3}}\right )}{12 x} \\ y(x)\to 0 \\ \end{align*}