78.5.3 problem 2 (c)
Internal
problem
ID
[18075]
Book
:
DIFFERENTIAL
EQUATIONS
WITH
APPLICATIONS
AND
HISTORICAL
NOTES
by
George
F.
Simmons.
3rd
edition.
2017.
CRC
press,
Boca
Raton
FL.
Section
:
Chapter
2.
First
order
equations.
Section
9
(Integrating
Factors).
Problems
at
page
80
Problem
number
:
2
(c)
Date
solved
:
Monday, March 31, 2025 at 05:06:52 PM
CAS
classification
:
[[_homogeneous, `class G`], _rational]
\begin{align*} x y^{\prime }+y+3 x^{3} y^{4} y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.052 (sec). Leaf size: 133
ode:=x*diff(y(x),x)+y(x)+3*x^3*y(x)^4*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= -\frac {\sqrt {6}\, \sqrt {\left (x -\sqrt {12 c_1^{2}+x^{2}}\right ) c_1 x}}{6 c_1 x} \\
y &= \frac {\sqrt {6}\, \sqrt {\left (x -\sqrt {12 c_1^{2}+x^{2}}\right ) c_1 x}}{6 c_1 x} \\
y &= -\frac {\sqrt {6}\, \sqrt {c_1 x \left (x +\sqrt {12 c_1^{2}+x^{2}}\right )}}{6 c_1 x} \\
y &= \frac {\sqrt {6}\, \sqrt {c_1 x \left (x +\sqrt {12 c_1^{2}+x^{2}}\right )}}{6 c_1 x} \\
\end{align*}
✓ Mathematica. Time used: 9.975 (sec). Leaf size: 166
ode=x*D[y[x],x]+y[x]+3*x^3*y[x]^4*D[y[x],x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to -\frac {\sqrt {c_1-\frac {\sqrt {x^2 \left (3+c_1{}^2 x^2\right )}}{x^2}}}{\sqrt {3}} \\
y(x)\to \frac {\sqrt {c_1-\frac {\sqrt {x^2 \left (3+c_1{}^2 x^2\right )}}{x^2}}}{\sqrt {3}} \\
y(x)\to -\frac {\sqrt {\frac {\sqrt {x^2 \left (3+c_1{}^2 x^2\right )}}{x^2}+c_1}}{\sqrt {3}} \\
y(x)\to \frac {\sqrt {\frac {\sqrt {x^2 \left (3+c_1{}^2 x^2\right )}}{x^2}+c_1}}{\sqrt {3}} \\
y(x)\to 0 \\
\end{align*}
✓ Sympy. Time used: 7.362 (sec). Leaf size: 126
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(3*x**3*y(x)**4*Derivative(y(x), x) + x*Derivative(y(x), x) + y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = - \frac {\sqrt {6} \sqrt {e^{C_{1}} - \frac {\sqrt {x^{2} e^{2 C_{1}} + 12}}{x}}}{6}, \ y{\left (x \right )} = \frac {\sqrt {6} \sqrt {e^{C_{1}} - \frac {\sqrt {x^{2} e^{2 C_{1}} + 12}}{x}}}{6}, \ y{\left (x \right )} = - \frac {\sqrt {6} \sqrt {e^{C_{1}} + \frac {\sqrt {x^{2} e^{2 C_{1}} + 12}}{x}}}{6}, \ y{\left (x \right )} = \frac {\sqrt {6} \sqrt {e^{C_{1}} + \frac {\sqrt {x^{2} e^{2 C_{1}} + 12}}{x}}}{6}\right ]
\]