Internal problem ID [6616]
Book: First order enumerated odes
Section: section 1
Problem number: 54.
ODE order: 1.
ODE degree: 3.
CAS Maple gives this as type [[_homogeneous, `class G`], _rational]
\[ \boxed {{y^{\prime }}^{3}-\frac {y^{2}}{x}=0} \]
✓ Solution by Maple
Time used: 0.094 (sec). Leaf size: 453
dsolve(diff(y(x),x)^3=y(x)^2/x,y(x), singsol=all)
\begin{align*} y \left (x \right ) = 0 \\ y \left (x \right ) = -\frac {3 x^{\frac {4}{3}} c_{1}}{8}+\frac {3 x^{\frac {2}{3}} c_{1}^{2}}{8}-\frac {c_{1}^{3}}{8}+\frac {x^{2}}{8} \\ y \left (x \right ) = -\frac {3 \left (-\frac {x^{\frac {2}{3}}}{2}-\frac {i \sqrt {3}\, x^{\frac {2}{3}}}{2}\right )^{2} c_{1}}{8}+\frac {3 \left (-\frac {x^{\frac {2}{3}}}{2}-\frac {i \sqrt {3}\, x^{\frac {2}{3}}}{2}\right ) c_{1}^{2}}{8}-\frac {c_{1}^{3}}{8}+\frac {x^{2}}{8} \\ y \left (x \right ) = -\frac {3 \left (-\frac {x^{\frac {2}{3}}}{2}+\frac {i \sqrt {3}\, x^{\frac {2}{3}}}{2}\right )^{2} c_{1}}{8}+\frac {3 \left (-\frac {x^{\frac {2}{3}}}{2}+\frac {i \sqrt {3}\, x^{\frac {2}{3}}}{2}\right ) c_{1}^{2}}{8}-\frac {c_{1}^{3}}{8}+\frac {x^{2}}{8} \\ y \left (x \right ) = \frac {3 \left (2 x^{\frac {2}{3}}+c_{1} \right )^{2} c_{1}}{64}-\frac {3 c_{1}^{2} \left (2 x^{\frac {2}{3}}+c_{1} \right )}{64}+\frac {c_{1}^{3}}{64}+\frac {x^{2}}{8} \\ y \left (x \right ) = \frac {3 \left (-x^{\frac {2}{3}}+c_{1} -i \sqrt {3}\, x^{\frac {2}{3}}\right )^{2} c_{1}}{64}-\frac {3 c_{1}^{2} \left (-x^{\frac {2}{3}}+c_{1} -i \sqrt {3}\, x^{\frac {2}{3}}\right )}{64}+\frac {c_{1}^{3}}{64}+\frac {x^{2}}{8} \\ y \left (x \right ) = \frac {3 \left (-x^{\frac {2}{3}}+c_{1} +i \sqrt {3}\, x^{\frac {2}{3}}\right )^{2} c_{1}}{64}-\frac {3 c_{1}^{2} \left (-x^{\frac {2}{3}}+c_{1} +i \sqrt {3}\, x^{\frac {2}{3}}\right )}{64}+\frac {c_{1}^{3}}{64}+\frac {x^{2}}{8} \\ y \left (x \right ) = -\frac {3 \left (2 x^{\frac {2}{3}}-c_{1} \right )^{2} c_{1}}{64}-\frac {3 c_{1}^{2} \left (2 x^{\frac {2}{3}}-c_{1} \right )}{64}-\frac {c_{1}^{3}}{64}+\frac {x^{2}}{8} \\ y \left (x \right ) = -\frac {3 \left (-x^{\frac {2}{3}}-c_{1} -i \sqrt {3}\, x^{\frac {2}{3}}\right )^{2} c_{1}}{64}-\frac {3 c_{1}^{2} \left (-x^{\frac {2}{3}}-c_{1} -i \sqrt {3}\, x^{\frac {2}{3}}\right )}{64}-\frac {c_{1}^{3}}{64}+\frac {x^{2}}{8} \\ y \left (x \right ) = -\frac {3 \left (-x^{\frac {2}{3}}-c_{1} +i \sqrt {3}\, x^{\frac {2}{3}}\right )^{2} c_{1}}{64}-\frac {3 c_{1}^{2} \left (-x^{\frac {2}{3}}-c_{1} +i \sqrt {3}\, x^{\frac {2}{3}}\right )}{64}-\frac {c_{1}^{3}}{64}+\frac {x^{2}}{8} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.083 (sec). Leaf size: 152
DSolve[(y'[x])^3==y[x]^2/x,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{216} \left (3 x^{2/3}+2 c_1\right ){}^3 \\ y(x)\to \frac {1}{216} \left (18 i \left (\sqrt {3}+i\right ) c_1{}^2 x^{2/3}-27 i \left (\sqrt {3}-i\right ) c_1 x^{4/3}+27 x^2+8 c_1{}^3\right ) \\ y(x)\to \frac {1}{216} \left (-18 i \left (\sqrt {3}-i\right ) c_1{}^2 x^{2/3}+27 i \left (\sqrt {3}+i\right ) c_1 x^{4/3}+27 x^2+8 c_1{}^3\right ) \\ y(x)\to 0 \\ \end{align*}