7.5.62 problem 62
Internal
problem
ID
[166]
Book
:
Elementary
Differential
Equations.
By
C.
Henry
Edwards,
David
E.
Penney
and
David
Calvis.
6th
edition.
2008
Section
:
Chapter
1.
First
order
differential
equations.
Section
1.6
(substitution
and
exact
equations).
Problems
at
page
72
Problem
number
:
62
Date
solved
:
Saturday, March 29, 2025 at 04:37:47 PM
CAS
classification
:
[[_homogeneous, `class A`], _rational, _dAlembert]
\begin{align*} y^{\prime }&=-\frac {y \left (2 x^{3}-y^{3}\right )}{x \left (2 y^{3}-x^{3}\right )} \end{align*}
✓ Maple. Time used: 0.010 (sec). Leaf size: 313
ode:=diff(y(x),x) = -y(x)*(2*x^3-y(x)^3)/x/(2*y(x)^3-x^3);
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {12^{{1}/{3}} \left (x 12^{{1}/{3}} c_1 +{\left (x \left (-9 c_1 \,x^{2}+\sqrt {3}\, \sqrt {\frac {27 c_1^{3} x^{4}-4 x}{c_1}}\right ) c_1^{2}\right )}^{{2}/{3}}\right )}{6 c_1 {\left (x \left (-9 c_1 \,x^{2}+\sqrt {3}\, \sqrt {\frac {27 c_1^{3} x^{4}-4 x}{c_1}}\right ) c_1^{2}\right )}^{{1}/{3}}} \\
y &= \frac {\left (\left (-i \sqrt {3}-1\right ) {\left (x \left (-9 c_1 \,x^{2}+\sqrt {3}\, \sqrt {\frac {27 c_1^{3} x^{4}-4 x}{c_1}}\right ) c_1^{2}\right )}^{{2}/{3}}+x \left (i 3^{{5}/{6}}-3^{{1}/{3}}\right ) c_1 2^{{2}/{3}}\right ) 3^{{1}/{3}} 2^{{2}/{3}}}{12 {\left (x \left (-9 c_1 \,x^{2}+\sqrt {3}\, \sqrt {\frac {27 c_1^{3} x^{4}-4 x}{c_1}}\right ) c_1^{2}\right )}^{{1}/{3}} c_1} \\
y &= -\frac {\left (\left (1-i \sqrt {3}\right ) {\left (x \left (-9 c_1 \,x^{2}+\sqrt {3}\, \sqrt {\frac {27 c_1^{3} x^{4}-4 x}{c_1}}\right ) c_1^{2}\right )}^{{2}/{3}}+x \left (i 3^{{5}/{6}}+3^{{1}/{3}}\right ) c_1 2^{{2}/{3}}\right ) 3^{{1}/{3}} 2^{{2}/{3}}}{12 {\left (x \left (-9 c_1 \,x^{2}+\sqrt {3}\, \sqrt {\frac {27 c_1^{3} x^{4}-4 x}{c_1}}\right ) c_1^{2}\right )}^{{1}/{3}} c_1} \\
\end{align*}
✓ Mathematica. Time used: 53.607 (sec). Leaf size: 440
ode=D[y[x],x]==- ( y[x]*(2*x^3-y[x]^3 ) )/( x*(2*y[x]^3-x^3) );
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \frac {\sqrt [3]{2} \left (-9 x^3+\sqrt {81 x^6-12 e^{3 c_1} x^3}\right ){}^{2/3}+2 \sqrt [3]{3} e^{c_1} x}{6^{2/3} \sqrt [3]{-9 x^3+\sqrt {81 x^6-12 e^{3 c_1} x^3}}} \\
y(x)\to \frac {i \sqrt [3]{2} \sqrt [6]{3} \left (\sqrt {3}+i\right ) \left (-9 x^3+\sqrt {81 x^6-12 e^{3 c_1} x^3}\right ){}^{2/3}-2 \left (\sqrt {3}+3 i\right ) e^{c_1} x}{2\ 2^{2/3} 3^{5/6} \sqrt [3]{-9 x^3+\sqrt {81 x^6-12 e^{3 c_1} x^3}}} \\
y(x)\to \frac {\sqrt [3]{2} \sqrt [6]{3} \left (-1-i \sqrt {3}\right ) \left (-9 x^3+\sqrt {81 x^6-12 e^{3 c_1} x^3}\right ){}^{2/3}-2 \left (\sqrt {3}-3 i\right ) e^{c_1} x}{2\ 2^{2/3} 3^{5/6} \sqrt [3]{-9 x^3+\sqrt {81 x^6-12 e^{3 c_1} x^3}}} \\
y(x)\to \frac {\sqrt [3]{\sqrt {x^6}-x^3}}{\sqrt [3]{2}} \\
y(x)\to -\frac {i \left (\sqrt {3}-i\right ) \sqrt [3]{\sqrt {x^6}-x^3}}{2 \sqrt [3]{2}} \\
y(x)\to \frac {i \left (\sqrt {3}+i\right ) \sqrt [3]{\sqrt {x^6}-x^3}}{2 \sqrt [3]{2}} \\
\end{align*}
✓ Sympy. Time used: 6.963 (sec). Leaf size: 5
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(Derivative(y(x), x) + (2*x**3 - y(x)**3)*y(x)/(x*(-x**3 + 2*y(x)**3)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
y{\left (x \right )} = C_{1} x
\]