50.8.14 problem 2(f)
Internal
problem
ID
[7930]
Book
:
Differential
Equations:
Theory,
Technique,
and
Practice
by
George
Simmons,
Steven
Krantz.
McGraw-Hill
NY.
2007.
1st
Edition.
Section
:
Chapter
1.
What
is
a
differential
equation.
Problems
for
Review
and
Discovery.
Page
53
Problem
number
:
2(f)
Date
solved
:
Sunday, March 30, 2025 at 12:38:18 PM
CAS
classification
:
[[_homogeneous, `class A`], _rational, _dAlembert]
\begin{align*} y^{\prime }&=\frac {x^{2}+2 y^{2}}{x^{2}-2 y^{2}} \end{align*}
✓ Maple. Time used: 0.004 (sec). Leaf size: 40
ode:=diff(y(x),x) = (x^2+2*y(x)^2)/(x^2-2*y(x)^2);
dsolve(ode,y(x), singsol=all);
\[
y = \operatorname {RootOf}\left (\int _{}^{\textit {\_Z}}\frac {2 \textit {\_a}^{2}-1}{2 \textit {\_a}^{3}+2 \textit {\_a}^{2}-\textit {\_a} +1}d \textit {\_a} +\ln \left (x \right )+c_1 \right ) x
\]
✓ Mathematica. Time used: 0.114 (sec). Leaf size: 80
ode=D[y[x],x]==(x^2+2*y[x]^2)/(x^2-2*y[x]^2);
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
\text {Solve}\left [\text {RootSum}\left [2 \text {$\#$1}^3+2 \text {$\#$1}^2-\text {$\#$1}+1\&,\frac {2 \text {$\#$1}^2 \log \left (\frac {y(x)}{x}-\text {$\#$1}\right )-\log \left (\frac {y(x)}{x}-\text {$\#$1}\right )}{6 \text {$\#$1}^2+4 \text {$\#$1}-1}\&\right ]=-\log (x)+c_1,y(x)\right ]
\]
✓ Sympy. Time used: 135.643 (sec). Leaf size: 345
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(Derivative(y(x), x) - (x**2 + 2*y(x)**2)/(x**2 - 2*y(x)**2),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
y{\left (x \right )} = \frac {C_{1} \left (\frac {x}{y{\left (x \right )}} - \frac {1}{3} + \frac {\sqrt [3]{5} \left (-1 - \sqrt {3} i\right ) \sqrt [3]{7 + 3 \sqrt {6}}}{6} - \frac {2 \cdot 5^{\frac {2}{3}}}{3 \left (-1 - \sqrt {3} i\right ) \sqrt [3]{7 + 3 \sqrt {6}}}\right )^{- \frac {1}{3} + \frac {2 \cdot 5^{\frac {2}{3}}}{15 \left (-1 - \sqrt {3} i\right ) \sqrt [3]{7 + 3 \sqrt {6}}} - \frac {\sqrt [3]{5} \left (-1 - \sqrt {3} i\right ) \sqrt [3]{7 + 3 \sqrt {6}}}{30}} \left (\frac {x}{y{\left (x \right )}} - \frac {1}{3} - \frac {2 \cdot 5^{\frac {2}{3}}}{3 \left (-1 + \sqrt {3} i\right ) \sqrt [3]{7 + 3 \sqrt {6}}} + \frac {\sqrt [3]{5} \left (-1 + \sqrt {3} i\right ) \sqrt [3]{7 + 3 \sqrt {6}}}{6}\right )^{- \frac {1}{3} - \frac {\sqrt [3]{5} \left (-1 + \sqrt {3} i\right ) \sqrt [3]{7 + 3 \sqrt {6}}}{30} + \frac {2 \cdot 5^{\frac {2}{3}}}{15 \left (-1 + \sqrt {3} i\right ) \sqrt [3]{7 + 3 \sqrt {6}}}}}{\left (\frac {x}{y{\left (x \right )}} - \frac {5^{\frac {2}{3}}}{3 \sqrt [3]{7 + 3 \sqrt {6}}} - \frac {1}{3} + \frac {\sqrt [3]{5} \sqrt [3]{7 + 3 \sqrt {6}}}{3}\right )^{- \frac {5^{\frac {2}{3}}}{15 \sqrt [3]{7 + 3 \sqrt {6}}} + \frac {\sqrt [3]{5} \sqrt [3]{7 + 3 \sqrt {6}}}{15} + \frac {1}{3}}}
\]