75.5.18 problem 117
Internal
problem
ID
[16683]
Book
:
A
book
of
problems
in
ordinary
differential
equations.
M.L.
KRASNOV,
A.L.
KISELYOV,
G.I.
MARKARENKO.
MIR,
MOSCOW.
1983
Section
:
Section
5.
Homogeneous
equations.
Exercises
page
44
Problem
number
:
117
Date
solved
:
Monday, March 31, 2025 at 03:05:23 PM
CAS
classification
:
[[_homogeneous, `class G`], _rational, _Bernoulli]
\begin{align*} 4 y^{6}+x^{3}&=6 x y^{5} y^{\prime } \end{align*}
✓ Maple. Time used: 0.003 (sec). Leaf size: 123
ode:=4*y(x)^6+x^3 = 6*x*y(x)^5*diff(y(x),x);
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \left (x^{3} \left (c_1 x -1\right )\right )^{{1}/{6}} \\
y &= -\left (x^{3} \left (c_1 x -1\right )\right )^{{1}/{6}} \\
y &= -\frac {\left (1+i \sqrt {3}\right ) \left (x^{3} \left (c_1 x -1\right )\right )^{{1}/{6}}}{2} \\
y &= \frac {\left (i \sqrt {3}-1\right ) \left (x^{3} \left (c_1 x -1\right )\right )^{{1}/{6}}}{2} \\
y &= -\frac {\left (i \sqrt {3}-1\right ) \left (x^{3} \left (c_1 x -1\right )\right )^{{1}/{6}}}{2} \\
y &= \frac {\left (1+i \sqrt {3}\right ) \left (x^{3} \left (c_1 x -1\right )\right )^{{1}/{6}}}{2} \\
\end{align*}
✓ Mathematica. Time used: 0.432 (sec). Leaf size: 144
ode=4*y[x]^6+x^3==6*x*y[x]^5*D[y[x],x];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to -\sqrt {x} \sqrt [6]{-1+c_1 x} \\
y(x)\to \sqrt {x} \sqrt [6]{-1+c_1 x} \\
y(x)\to -\sqrt [3]{-1} \sqrt {x} \sqrt [6]{-1+c_1 x} \\
y(x)\to \sqrt [3]{-1} \sqrt {x} \sqrt [6]{-1+c_1 x} \\
y(x)\to -(-1)^{2/3} \sqrt {x} \sqrt [6]{-1+c_1 x} \\
y(x)\to (-1)^{2/3} \sqrt {x} \sqrt [6]{-1+c_1 x} \\
\end{align*}
✓ Sympy. Time used: 4.848 (sec). Leaf size: 126
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x**3 - 6*x*y(x)**5*Derivative(y(x), x) + 4*y(x)**6,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = - \sqrt [6]{x^{3} \left (C_{1} x - 1\right )}, \ y{\left (x \right )} = \sqrt [6]{x^{3} \left (C_{1} x - 1\right )}, \ y{\left (x \right )} = \frac {\sqrt [6]{x^{3} \left (C_{1} x - 1\right )} \left (-1 - \sqrt {3} i\right )}{2}, \ y{\left (x \right )} = \frac {\sqrt [6]{x^{3} \left (C_{1} x - 1\right )} \left (-1 + \sqrt {3} i\right )}{2}, \ y{\left (x \right )} = \frac {\sqrt [6]{x^{3} \left (C_{1} x - 1\right )} \left (1 - \sqrt {3} i\right )}{2}, \ y{\left (x \right )} = \frac {\sqrt [6]{x^{3} \left (C_{1} x - 1\right )} \left (1 + \sqrt {3} i\right )}{2}\right ]
\]