85.18.8 problem 1 (h)
Internal
problem
ID
[22554]
Book
:
Applied
Differential
Equations.
By
Murray
R.
Spiegel.
3rd
edition.
1980.
Pearson.
ISBN
978-0130400970
Section
:
Chapter
two.
First
order
and
simple
higher
order
ordinary
differential
equations.
A
Exercises
at
page
52
Problem
number
:
1
(h)
Date
solved
:
Thursday, October 02, 2025 at 08:50:34 PM
CAS
classification
:
[[_homogeneous, `class G`], _rational]
\begin{align*} y^{\prime }&=\frac {y}{y^{3}-3 x} \end{align*}
✓ Maple. Time used: 0.006 (sec). Leaf size: 167
ode:=diff(y(x),x) = y(x)/(y(x)^3-3*x);
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \left (3 x -\sqrt {9 x^{2}-6 c_1}\right )^{{1}/{3}} \\
y &= \left (3 x +\sqrt {9 x^{2}-6 c_1}\right )^{{1}/{3}} \\
y &= -\frac {\left (3 x -\sqrt {9 x^{2}-6 c_1}\right )^{{1}/{3}} \left (1+i \sqrt {3}\right )}{2} \\
y &= \frac {\left (3 x -\sqrt {9 x^{2}-6 c_1}\right )^{{1}/{3}} \left (i \sqrt {3}-1\right )}{2} \\
y &= -\frac {\left (3 x +\sqrt {9 x^{2}-6 c_1}\right )^{{1}/{3}} \left (1+i \sqrt {3}\right )}{2} \\
y &= \frac {\left (3 x +\sqrt {9 x^{2}-6 c_1}\right )^{{1}/{3}} \left (i \sqrt {3}-1\right )}{2} \\
\end{align*}
✓ Mathematica. Time used: 2.019 (sec). Leaf size: 194
ode=D[y[x],x]== y[x]/( y[x]^3-3*x);
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \sqrt [3]{3 x-\sqrt {9 x^2-6 c_1}}\\ y(x)&\to -\sqrt [3]{-1} \sqrt [3]{3 x-\sqrt {9 x^2-6 c_1}}\\ y(x)&\to (-1)^{2/3} \sqrt [3]{3 x-\sqrt {9 x^2-6 c_1}}\\ y(x)&\to \sqrt [3]{3 x+\sqrt {9 x^2-6 c_1}}\\ y(x)&\to -\sqrt [3]{-1} \sqrt [3]{3 x+\sqrt {9 x^2-6 c_1}}\\ y(x)&\to (-1)^{2/3} \sqrt [3]{3 x+\sqrt {9 x^2-6 c_1}}\\ y(x)&\to 0 \end{align*}
✓ Sympy. Time used: 7.547 (sec). Leaf size: 187
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(Derivative(y(x), x) - y(x)/(-3*x + y(x)**3),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = \sqrt [3]{3 x - \sqrt {3} \sqrt {C_{1} + 3 x^{2}}}, \ y{\left (x \right )} = \sqrt [3]{3 x + \sqrt {3} \sqrt {C_{1} + 3 x^{2}}}, \ y{\left (x \right )} = \frac {\left (-1 - \sqrt {3} i\right ) \sqrt [3]{3 x - \sqrt {3} \sqrt {C_{1} + 3 x^{2}}}}{2}, \ y{\left (x \right )} = \frac {\left (-1 + \sqrt {3} i\right ) \sqrt [3]{3 x - \sqrt {3} \sqrt {C_{1} + 3 x^{2}}}}{2}, \ y{\left (x \right )} = \frac {\left (-1 - \sqrt {3} i\right ) \sqrt [3]{3 x + \sqrt {3} \sqrt {C_{1} + 3 x^{2}}}}{2}, \ y{\left (x \right )} = \frac {\left (-1 + \sqrt {3} i\right ) \sqrt [3]{3 x + \sqrt {3} \sqrt {C_{1} + 3 x^{2}}}}{2}\right ]
\]