57.1.59 problem 59
Internal
problem
ID
[9043]
Book
:
First
order
enumerated
odes
Section
:
section
1
Problem
number
:
59
Date
solved
:
Wednesday, March 05, 2025 at 07:15:18 AM
CAS
classification
:
[_separable]
\begin{align*} {y^{\prime }}^{2}&=\frac {1}{x^{3} y^{4}} \end{align*}
✓ Maple. Time used: 0.051 (sec). Leaf size: 133
ode:=diff(y(x),x)^2 = 1/x^3/y(x)^4;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \left (\frac {\sqrt {x}\, c_{1} -6}{\sqrt {x}}\right )^{{1}/{3}} \\
y &= -\frac {\left (\frac {\sqrt {x}\, c_{1} -6}{\sqrt {x}}\right )^{{1}/{3}} \left (1+i \sqrt {3}\right )}{2} \\
y &= \frac {\left (\frac {\sqrt {x}\, c_{1} -6}{\sqrt {x}}\right )^{{1}/{3}} \left (i \sqrt {3}-1\right )}{2} \\
y &= \left (\frac {\sqrt {x}\, c_{1} +6}{\sqrt {x}}\right )^{{1}/{3}} \\
y &= -\frac {\left (\frac {\sqrt {x}\, c_{1} +6}{\sqrt {x}}\right )^{{1}/{3}} \left (1+i \sqrt {3}\right )}{2} \\
y &= \frac {\left (\frac {\sqrt {x}\, c_{1} +6}{\sqrt {x}}\right )^{{1}/{3}} \left (i \sqrt {3}-1\right )}{2} \\
\end{align*}
✓ Mathematica. Time used: 3.402 (sec). Leaf size: 157
ode=(D[y[x],x])^2==1/(x^3*y[x]^4);
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to -\sqrt [3]{-3} \sqrt [3]{-\frac {2}{\sqrt {x}}+c_1} \\
y(x)\to \sqrt [3]{3} \sqrt [3]{-\frac {2}{\sqrt {x}}+c_1} \\
y(x)\to (-1)^{2/3} \sqrt [3]{3} \sqrt [3]{-\frac {2}{\sqrt {x}}+c_1} \\
y(x)\to -\sqrt [3]{-3} \sqrt [3]{\frac {2}{\sqrt {x}}+c_1} \\
y(x)\to \sqrt [3]{3} \sqrt [3]{\frac {2}{\sqrt {x}}+c_1} \\
y(x)\to (-1)^{2/3} \sqrt [3]{3} \sqrt [3]{\frac {2}{\sqrt {x}}+c_1} \\
\end{align*}
✓ Sympy. Time used: 4.459 (sec). Leaf size: 170
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(Derivative(y(x), x)**2 - 1/(x**3*y(x)**4),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = \sqrt [3]{C_{1} + 6 x \sqrt {\frac {1}{x^{3}}}}, \ y{\left (x \right )} = \frac {\left (- \sqrt [3]{3} - 3^{\frac {5}{6}} i\right ) \sqrt [3]{C_{1} + 2 x \sqrt {\frac {1}{x^{3}}}}}{2}, \ y{\left (x \right )} = \frac {\left (- \sqrt [3]{3} + 3^{\frac {5}{6}} i\right ) \sqrt [3]{C_{1} + 2 x \sqrt {\frac {1}{x^{3}}}}}{2}, \ y{\left (x \right )} = \sqrt [3]{C_{1} - 6 x \sqrt {\frac {1}{x^{3}}}}, \ y{\left (x \right )} = \frac {\left (- \sqrt [3]{3} - 3^{\frac {5}{6}} i\right ) \sqrt [3]{C_{1} - 2 x \sqrt {\frac {1}{x^{3}}}}}{2}, \ y{\left (x \right )} = \frac {\left (- \sqrt [3]{3} + 3^{\frac {5}{6}} i\right ) \sqrt [3]{C_{1} - 2 x \sqrt {\frac {1}{x^{3}}}}}{2}\right ]
\]