80.3.6 problem 6
Internal
problem
ID
[18479]
Book
:
Elementary
Differential
Equations.
By
Thornton
C.
Fry.
D
Van
Nostrand.
NY.
First
Edition
(1929)
Section
:
Chapter
IV.
Methods
of
solution:
First
order
equations.
section
29.
Problems
at
page
81
Problem
number
:
6
Date
solved
:
Monday, March 31, 2025 at 05:36:16 PM
CAS
classification
:
[[_homogeneous, `class A`], _exact, _rational, _dAlembert]
\begin{align*} \frac {2 x}{y^{3}}+\left (\frac {1}{y^{2}}-\frac {3 x^{2}}{y^{4}}\right ) y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.010 (sec). Leaf size: 313
ode:=2*x/y(x)^3+(1/y(x)^2-3*x^2/y(x)^4)*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {1+\frac {\left (12 \sqrt {3}\, x \sqrt {27 x^{2} c_1^{2}-4}\, c_1 -108 x^{2} c_1^{2}+8\right )^{{1}/{3}}}{2}+\frac {2}{\left (12 \sqrt {3}\, x \sqrt {27 x^{2} c_1^{2}-4}\, c_1 -108 x^{2} c_1^{2}+8\right )^{{1}/{3}}}}{3 c_1} \\
y &= -\frac {\left (1+i \sqrt {3}\right ) \left (12 \sqrt {3}\, x \sqrt {27 x^{2} c_1^{2}-4}\, c_1 -108 x^{2} c_1^{2}+8\right )^{{2}/{3}}-4 i \sqrt {3}-4 \left (12 \sqrt {3}\, x \sqrt {27 x^{2} c_1^{2}-4}\, c_1 -108 x^{2} c_1^{2}+8\right )^{{1}/{3}}+4}{12 \left (12 \sqrt {3}\, x \sqrt {27 x^{2} c_1^{2}-4}\, c_1 -108 x^{2} c_1^{2}+8\right )^{{1}/{3}} c_1} \\
y &= \frac {\left (i \sqrt {3}-1\right ) \left (12 \sqrt {3}\, x \sqrt {27 x^{2} c_1^{2}-4}\, c_1 -108 x^{2} c_1^{2}+8\right )^{{2}/{3}}-4 i \sqrt {3}+4 \left (12 \sqrt {3}\, x \sqrt {27 x^{2} c_1^{2}-4}\, c_1 -108 x^{2} c_1^{2}+8\right )^{{1}/{3}}-4}{12 \left (12 \sqrt {3}\, x \sqrt {27 x^{2} c_1^{2}-4}\, c_1 -108 x^{2} c_1^{2}+8\right )^{{1}/{3}} c_1} \\
\end{align*}
✓ Mathematica. Time used: 60.18 (sec). Leaf size: 458
ode=2*x/y[x]^3+(1/y[x]^2-3*x^2/y[x]^4)*D[y[x],x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \frac {1}{3} \left (\frac {\sqrt [3]{27 e^{c_1} x^2+3 \sqrt {81 e^{2 c_1} x^4-12 e^{4 c_1} x^2}-2 e^{3 c_1}}}{\sqrt [3]{2}}+\frac {\sqrt [3]{2} e^{2 c_1}}{\sqrt [3]{27 e^{c_1} x^2+3 \sqrt {81 e^{2 c_1} x^4-12 e^{4 c_1} x^2}-2 e^{3 c_1}}}-e^{c_1}\right ) \\
y(x)\to \frac {i \left (\sqrt {3}+i\right ) \sqrt [3]{27 e^{c_1} x^2+3 \sqrt {81 e^{2 c_1} x^4-12 e^{4 c_1} x^2}-2 e^{3 c_1}}}{6 \sqrt [3]{2}}-\frac {i \left (\sqrt {3}-i\right ) e^{2 c_1}}{3\ 2^{2/3} \sqrt [3]{27 e^{c_1} x^2+3 \sqrt {81 e^{2 c_1} x^4-12 e^{4 c_1} x^2}-2 e^{3 c_1}}}-\frac {e^{c_1}}{3} \\
y(x)\to -\frac {i \left (\sqrt {3}-i\right ) \sqrt [3]{27 e^{c_1} x^2+3 \sqrt {81 e^{2 c_1} x^4-12 e^{4 c_1} x^2}-2 e^{3 c_1}}}{6 \sqrt [3]{2}}+\frac {i \left (\sqrt {3}+i\right ) e^{2 c_1}}{3\ 2^{2/3} \sqrt [3]{27 e^{c_1} x^2+3 \sqrt {81 e^{2 c_1} x^4-12 e^{4 c_1} x^2}-2 e^{3 c_1}}}-\frac {e^{c_1}}{3} \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(2*x/y(x)**3 + (-3*x**2/y(x)**4 + y(x)**(-2))*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out