50.6.3 problem 1(c)
Internal
problem
ID
[7895]
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
1.8.
Integrating
Factors.
Page
32
Problem
number
:
1(c)
Date
solved
:
Wednesday, March 05, 2025 at 05:16:47 AM
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: 0.099 (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 {x c_{1} \left (x -\sqrt {12 c_{1}^{2}+x^{2}}\right )}}{6 x c_{1}} \\
y &= \frac {\sqrt {6}\, \sqrt {x c_{1} \left (x -\sqrt {12 c_{1}^{2}+x^{2}}\right )}}{6 x c_{1}} \\
y &= -\frac {\sqrt {6}\, \sqrt {x c_{1} \left (x +\sqrt {12 c_{1}^{2}+x^{2}}\right )}}{6 x c_{1}} \\
y &= \frac {\sqrt {6}\, \sqrt {x c_{1} \left (x +\sqrt {12 c_{1}^{2}+x^{2}}\right )}}{6 x c_{1}} \\
\end{align*}
✓ Mathematica. Time used: 10.211 (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: 7.683 (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)
\[
\left [ y{\left (x \right )} = - \frac {\sqrt {6} \sqrt {e^{C_{1}} - \frac {\sqrt {x^{2} e^{2 C_{1}} + 12}}{x}}}{6}, \ y{\left (x \right )} = \frac {\sqrt {6} \sqrt {e^{C_{1}} - \frac {\sqrt {x^{2} e^{2 C_{1}} + 12}}{x}}}{6}, \ y{\left (x \right )} = - \frac {\sqrt {6} \sqrt {e^{C_{1}} + \frac {\sqrt {x^{2} e^{2 C_{1}} + 12}}{x}}}{6}, \ y{\left (x \right )} = \frac {\sqrt {6} \sqrt {e^{C_{1}} + \frac {\sqrt {x^{2} e^{2 C_{1}} + 12}}{x}}}{6}\right ]
\]