Internal problem ID [6617]
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]
Solve \begin {gather*} \boxed {\left (y^{\prime }\right )^{3}-\frac {y^{2}}{x}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.205 (sec). Leaf size: 311
dsolve(diff(y(x),x)^3=y(x)^2/x,y(x), singsol=all)
\begin{align*} y \relax (x ) = 0 \\ y \relax (x ) = -\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 \relax (x ) = -\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 \relax (x ) = -\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} \\ -\frac {3 i \ln \left (\left (x^{2} y \relax (x )^{2}\right )^{\frac {1}{3}} \left (1+i \sqrt {3}\right )+4 y \relax (x )\right ) \sqrt {3}}{2 \left (1+i \sqrt {3}\right )}-\frac {3 \ln \left (\left (x^{2} y \relax (x )^{2}\right )^{\frac {1}{3}} \left (1+i \sqrt {3}\right )+4 y \relax (x )\right )}{2 \left (1+i \sqrt {3}\right )}+\ln \left (y \relax (x )\right )+c_{1} = 0 \\ -\frac {3 i \ln \left (\left (x^{2} y \relax (x )^{2}\right )^{\frac {1}{3}} \left (-1+i \sqrt {3}\right )-4 y \relax (x )\right ) \sqrt {3}}{2 \left (-1+i \sqrt {3}\right )}+\frac {3 \ln \left (\left (x^{2} y \relax (x )^{2}\right )^{\frac {1}{3}} \left (-1+i \sqrt {3}\right )-4 y \relax (x )\right )}{2 \left (-1+i \sqrt {3}\right )}+\ln \left (y \relax (x )\right )+c_{1} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.08 (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*}