50.4.21 problem 21
Internal
problem
ID
[7870]
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.
Section
1.5.
Exact
Equations.
Page
20
Problem
number
:
21
Date
solved
:
Sunday, March 30, 2025 at 12:33:47 PM
CAS
classification
:
[[_homogeneous, `class A`], _exact, _rational, _dAlembert]
\begin{align*} \frac {4 y^{2}-2 x^{2}}{4 x y^{2}-x^{3}}+\frac {\left (8 y^{2}-x^{2}\right ) y^{\prime }}{4 y^{3}-x^{2} y}&=0 \end{align*}
✓ Maple. Time used: 0.136 (sec). Leaf size: 71
ode:=(4*y(x)^2-2*x^2)/(4*x*y(x)^2-x^3)+(8*y(x)^2-x^2)/(4*y(x)^3-x^2*y(x))*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {\sqrt {\frac {2 c_1 \,x^{3}-2 \sqrt {c_1^{2} x^{6}+16}}{c_1 \,x^{3}}}\, x}{4} \\
y &= \frac {\sqrt {2}\, \sqrt {\frac {c_1 \,x^{3}+\sqrt {c_1^{2} x^{6}+16}}{c_1 \,x^{3}}}\, x}{4} \\
\end{align*}
✓ Mathematica. Time used: 12.795 (sec). Leaf size: 297
ode=( (4*y[x]^2-2*x^2)/(4*x*y[x]^2-x^3))+( (8*y[x]^2-x^2)/(4*y[x]^3-x^2*y[x]) )*D[y[x],x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to -\frac {\sqrt {x^2-\frac {\sqrt {x^6-16 e^{2 c_1}}}{x}}}{2 \sqrt {2}} \\
y(x)\to \frac {\sqrt {x^2-\frac {\sqrt {x^6-16 e^{2 c_1}}}{x}}}{2 \sqrt {2}} \\
y(x)\to -\frac {\sqrt {\frac {x^3+\sqrt {x^6-16 e^{2 c_1}}}{x}}}{2 \sqrt {2}} \\
y(x)\to \frac {\sqrt {\frac {x^3+\sqrt {x^6-16 e^{2 c_1}}}{x}}}{2 \sqrt {2}} \\
y(x)\to -\frac {\sqrt {x^2-\frac {\sqrt {x^6}}{x}}}{2 \sqrt {2}} \\
y(x)\to \frac {\sqrt {x^2-\frac {\sqrt {x^6}}{x}}}{2 \sqrt {2}} \\
y(x)\to -\frac {\sqrt {\frac {\sqrt {x^6}+x^3}{x}}}{2 \sqrt {2}} \\
y(x)\to \frac {\sqrt {\frac {\sqrt {x^6}+x^3}{x}}}{2 \sqrt {2}} \\
\end{align*}
✓ Sympy. Time used: 5.901 (sec). Leaf size: 105
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq((-2*x**2 + 4*y(x)**2)/(-x**3 + 4*x*y(x)**2) + (-x**2 + 8*y(x)**2)*Derivative(y(x), x)/(-x**2*y(x) + 4*y(x)**3),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {x^{2} - \frac {\sqrt {C_{1} + x^{6}}}{x}}}{4}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {x^{2} - \frac {\sqrt {C_{1} + x^{6}}}{x}}}{4}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {x^{2} + \frac {\sqrt {C_{1} + x^{6}}}{x}}}{4}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {x^{2} + \frac {\sqrt {C_{1} + x^{6}}}{x}}}{4}\right ]
\]