Internal problem ID [2591]
Book: Differential equations with applications and historial notes, George F. Simmons,
1971
Section: Chapter 2, section 10, page 47
Problem number: 2(a).
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, class A], _rational, _dAlembert]
Solve \begin {gather*} \boxed {\left (3 x^{2}-y^{2}\right ) y^{\prime }-2 y x=0} \end {gather*}
✓ Solution by Maple
Time used: 0.021 (sec). Leaf size: 402
dsolve((3*x^2-y(x)^2)*diff(y(x),x)-2*x*y(x)=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\left (12 \sqrt {3}\, x \sqrt {27 c_{1}^{2} x^{2}-4}\, c_{1}-108 c_{1}^{2} x^{2}+8\right )^{\frac {1}{3}}}{6 c_{1}}+\frac {2}{3 c_{1} \left (12 \sqrt {3}\, x \sqrt {27 c_{1}^{2} x^{2}-4}\, c_{1}-108 c_{1}^{2} x^{2}+8\right )^{\frac {1}{3}}}+\frac {1}{3 c_{1}} \\ y \relax (x ) = -\frac {\left (12 \sqrt {3}\, x \sqrt {27 c_{1}^{2} x^{2}-4}\, c_{1}-108 c_{1}^{2} x^{2}+8\right )^{\frac {1}{3}}}{12 c_{1}}-\frac {1}{3 c_{1} \left (12 \sqrt {3}\, x \sqrt {27 c_{1}^{2} x^{2}-4}\, c_{1}-108 c_{1}^{2} x^{2}+8\right )^{\frac {1}{3}}}+\frac {1}{3 c_{1}}-\frac {i \sqrt {3}\, \left (\frac {\left (12 \sqrt {3}\, x \sqrt {27 c_{1}^{2} x^{2}-4}\, c_{1}-108 c_{1}^{2} x^{2}+8\right )^{\frac {1}{3}}}{6 c_{1}}-\frac {2}{3 c_{1} \left (12 \sqrt {3}\, x \sqrt {27 c_{1}^{2} x^{2}-4}\, c_{1}-108 c_{1}^{2} x^{2}+8\right )^{\frac {1}{3}}}\right )}{2} \\ y \relax (x ) = -\frac {\left (12 \sqrt {3}\, x \sqrt {27 c_{1}^{2} x^{2}-4}\, c_{1}-108 c_{1}^{2} x^{2}+8\right )^{\frac {1}{3}}}{12 c_{1}}-\frac {1}{3 c_{1} \left (12 \sqrt {3}\, x \sqrt {27 c_{1}^{2} x^{2}-4}\, c_{1}-108 c_{1}^{2} x^{2}+8\right )^{\frac {1}{3}}}+\frac {1}{3 c_{1}}+\frac {i \sqrt {3}\, \left (\frac {\left (12 \sqrt {3}\, x \sqrt {27 c_{1}^{2} x^{2}-4}\, c_{1}-108 c_{1}^{2} x^{2}+8\right )^{\frac {1}{3}}}{6 c_{1}}-\frac {2}{3 c_{1} \left (12 \sqrt {3}\, x \sqrt {27 c_{1}^{2} x^{2}-4}\, c_{1}-108 c_{1}^{2} x^{2}+8\right )^{\frac {1}{3}}}\right )}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 33.173 (sec). Leaf size: 419
DSolve[(3*x^2-y[x]^2)*y'[x]-2*x*y[x]==0,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*}