Internal problem ID [5013]
Book: Ordinary differential equations and calculus of variations. Makarets and Reshetnyak. Wold
Scientific. Singapore. 1995
Section: Chapter 1. First order differential equations. Section 1.2 Homogeneous equations problems.
page 12
Problem number: 18.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, class A], _rational, _dAlembert]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {2 x y}{3 x^{2}-y^{2}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.026 (sec). Leaf size: 402
dsolve(diff(y(x),x)=2*x*y(x)/(3*x^2-y(x)^2),y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\left (-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, x \sqrt {27 c_{1}^{2} x^{2}-4}\, c_{1}+8\right )^{\frac {1}{3}}}{6 c_{1}}+\frac {2}{3 c_{1} \left (-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, x \sqrt {27 c_{1}^{2} x^{2}-4}\, c_{1}+8\right )^{\frac {1}{3}}}+\frac {1}{3 c_{1}} \\ y \relax (x ) = -\frac {\left (-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, x \sqrt {27 c_{1}^{2} x^{2}-4}\, c_{1}+8\right )^{\frac {1}{3}}}{12 c_{1}}-\frac {1}{3 c_{1} \left (-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, x \sqrt {27 c_{1}^{2} x^{2}-4}\, c_{1}+8\right )^{\frac {1}{3}}}+\frac {1}{3 c_{1}}-\frac {i \sqrt {3}\, \left (\frac {\left (-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, x \sqrt {27 c_{1}^{2} x^{2}-4}\, c_{1}+8\right )^{\frac {1}{3}}}{6 c_{1}}-\frac {2}{3 c_{1} \left (-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, x \sqrt {27 c_{1}^{2} x^{2}-4}\, c_{1}+8\right )^{\frac {1}{3}}}\right )}{2} \\ y \relax (x ) = -\frac {\left (-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, x \sqrt {27 c_{1}^{2} x^{2}-4}\, c_{1}+8\right )^{\frac {1}{3}}}{12 c_{1}}-\frac {1}{3 c_{1} \left (-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, x \sqrt {27 c_{1}^{2} x^{2}-4}\, c_{1}+8\right )^{\frac {1}{3}}}+\frac {1}{3 c_{1}}+\frac {i \sqrt {3}\, \left (\frac {\left (-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, x \sqrt {27 c_{1}^{2} x^{2}-4}\, c_{1}+8\right )^{\frac {1}{3}}}{6 c_{1}}-\frac {2}{3 c_{1} \left (-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, x \sqrt {27 c_{1}^{2} x^{2}-4}\, c_{1}+8\right )^{\frac {1}{3}}}\right )}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 39.537 (sec). Leaf size: 419
DSolve[y'[x]==2*x*y[x]/(3*x^2-y[x]^2),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{3} \left (e^{c_1} \left (-1+\frac {\sqrt [3]{2} e^{c_1}}{\sqrt [3]{27 e^{c_1} x^2+3 \sqrt {81 e^{2 c_1} x^4-12 e^{4 c_1} x^2}-2 e^{3 c_1}}}\right )+\frac {\sqrt [3]{27 e^{c_1} x^2+3 \sqrt {81 e^{2 c_1} x^4-12 e^{4 c_1} x^2}-2 e^{3 c_1}}}{\sqrt [3]{2}}\right ) \\ y(x)\to \frac {1}{6} \left ((-2)^{2/3} \sqrt [3]{27 e^{c_1} x^2+3 \sqrt {81 e^{2 c_1} x^4-12 e^{4 c_1} x^2}-2 e^{3 c_1}}-\frac {2 \sqrt [3]{-2} e^{2 c_1}}{\sqrt [3]{27 e^{c_1} x^2+3 \sqrt {81 e^{2 c_1} x^4-12 e^{4 c_1} x^2}-2 e^{3 c_1}}}-2 e^{c_1}\right ) \\ y(x)\to \frac {1}{3} e^{c_1} \left (-1+\frac {e^{c_1} \text {Root}\left [\text {$\#$1}^3-2\&,3\right ]}{\sqrt [3]{27 e^{c_1} x^2+3 \sqrt {81 e^{2 c_1} x^4-12 e^{4 c_1} x^2}-2 e^{3 c_1}}}\right )-\frac {1}{3} \sqrt [3]{-\frac {1}{2}} \sqrt [3]{27 e^{c_1} x^2+3 \sqrt {81 e^{2 c_1} x^4-12 e^{4 c_1} x^2}-2 e^{3 c_1}} \\ y(x)\to 0 \\ \end{align*}