57.1.54 problem 54
Internal
problem
ID
[9038]
Book
:
First
order
enumerated
odes
Section
:
section
1
Problem
number
:
54
Date
solved
:
Sunday, March 30, 2025 at 01:59:55 PM
CAS
classification
:
[[_homogeneous, `class G`], _rational]
\begin{align*} {y^{\prime }}^{3}&=\frac {y^{2}}{x} \end{align*}
✓ Maple. Time used: 0.925 (sec). Leaf size: 341
ode:=diff(y(x),x)^3 = y(x)^2/x;
dsolve(ode,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 (-1-i \sqrt {3}\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 (-1-i \sqrt {3}\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 (1+i \sqrt {3}\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 (-1-i \sqrt {3}\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*}
✓ Mathematica. Time used: 0.088 (sec). Leaf size: 152
ode=(D[y[x],x])^3==y[x]^2/x;
ic={};
DSolve[{ode,ic},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*}
✓ Sympy. Time used: 2.293 (sec). Leaf size: 109
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(Derivative(y(x), x)**3 - y(x)**2/x,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ - \frac {3 x \sqrt [3]{\frac {y^{2}{\left (x \right )}}{x}}}{2 y^{\frac {2}{3}}{\left (x \right )}} + 3 \sqrt [3]{y{\left (x \right )}} = C_{1}, \ \frac {3 x \sqrt [3]{\frac {y^{2}{\left (x \right )}}{x}} \left (1 + \sqrt {3} i\right )}{4 y^{\frac {2}{3}}{\left (x \right )}} + 3 \sqrt [3]{y{\left (x \right )}} = C_{1}, \ \frac {3 x \sqrt [3]{\frac {y^{2}{\left (x \right )}}{x}} \left (1 - \sqrt {3} i\right )}{4 y^{\frac {2}{3}}{\left (x \right )}} + 3 \sqrt [3]{y{\left (x \right )}} = C_{1}\right ]
\]