57.1.54 problem 54

Internal problem ID [9038]
Book : First order enumerated odes
Section : section 1
Problem number : 54
Date solved : Monday, January 27, 2025 at 05:27:48 PM
CAS classification : [[_homogeneous, `class G`], _rational]

\begin{align*} {y^{\prime }}^{3}&=\frac {y^{2}}{x} \end{align*}

✓ Solution by Maple

Time used: 0.092 (sec). Leaf size: 341

dsolve(diff(y(x),x)^3=y(x)^2/x,y(x), singsol=all)
 

\begin{align*} y &= 0 \\ y &= -\frac {3 x^{{4}/{3}} c_{1}}{8}+\frac {3 x^{{2}/{3}} c_{1}^{2}}{8}-\frac {c_{1}^{3}}{8}+\frac {x^{2}}{8} \\ y &= \frac {3 \left (-i \sqrt {3}-1\right ) c_{1}^{2} x^{{2}/{3}}}{16}+\frac {3 c_{1} \left (1-i \sqrt {3}\right ) x^{{4}/{3}}}{16}-\frac {c_{1}^{3}}{8}+\frac {x^{2}}{8} \\ y &= \frac {3 \left (i \sqrt {3}-1\right ) c_{1}^{2} x^{{2}/{3}}}{16}+\frac {3 c_{1} \left (1+i \sqrt {3}\right ) x^{{4}/{3}}}{16}-\frac {c_{1}^{3}}{8}+\frac {x^{2}}{8} \\ y &= \frac {3 x^{{4}/{3}} c_{1}}{16}+\frac {3 x^{{2}/{3}} c_{1}^{2}}{32}+\frac {c_{1}^{3}}{64}+\frac {x^{2}}{8} \\ y &= \frac {3 \left (-i \sqrt {3}-1\right ) c_{1}^{2} x^{{2}/{3}}}{64}+\frac {3 \left (i \sqrt {3}-1\right ) c_{1} x^{{4}/{3}}}{32}+\frac {c_{1}^{3}}{64}+\frac {x^{2}}{8} \\ y &= \frac {3 \left (i \sqrt {3}-1\right ) c_{1}^{2} x^{{2}/{3}}}{64}+\frac {3 c_{1} \left (-i \sqrt {3}-1\right ) x^{{4}/{3}}}{32}+\frac {c_{1}^{3}}{64}+\frac {x^{2}}{8} \\ y &= -\frac {3 x^{{4}/{3}} c_{1}}{16}+\frac {3 x^{{2}/{3}} c_{1}^{2}}{32}-\frac {c_{1}^{3}}{64}+\frac {x^{2}}{8} \\ y &= \frac {3 \left (-i \sqrt {3}-1\right ) c_{1}^{2} x^{{2}/{3}}}{64}+\frac {3 c_{1} \left (1-i \sqrt {3}\right ) x^{{4}/{3}}}{32}-\frac {c_{1}^{3}}{64}+\frac {x^{2}}{8} \\ y &= \frac {3 \left (i \sqrt {3}-1\right ) c_{1}^{2} x^{{2}/{3}}}{64}+\frac {3 c_{1} \left (1+i \sqrt {3}\right ) x^{{4}/{3}}}{32}-\frac {c_{1}^{3}}{64}+\frac {x^{2}}{8} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.088 (sec). Leaf size: 152

DSolve[(D[y[x],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*}