67.6.19 problem 7.5 (i)
Internal
problem
ID
[16445]
Book
:
Ordinary
Differential
Equations.
An
introduction
to
the
fundamentals.
Kenneth
B.
Howell.
second
edition.
CRC
Press.
FL,
USA.
2020
Section
:
Chapter
7.
The
exact
form
and
general
integrating
fators.
Additional
exercises.
page
141
Problem
number
:
7.5
(i)
Date
solved
:
Thursday, October 02, 2025 at 01:33:25 PM
CAS
classification
:
[[_homogeneous, `class G`], _rational]
\begin{align*} 6+12 x^{2} y^{2}+\left (7 x^{3} y+\frac {x}{y}\right ) y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.231 (sec). Leaf size: 63
ode:=6+12*x^2*y(x)^2+(7*x^3*y(x)+x/y(x))*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\[
y = \operatorname {RootOf}\left (x^{10} \textit {\_Z}^{35} c_1^{2}-x^{10} \textit {\_Z}^{30} c_1^{2}-1\right )^{15} x^{4} \left (\operatorname {RootOf}\left (x^{10} \textit {\_Z}^{35} c_1^{2}-x^{10} \textit {\_Z}^{30} c_1^{2}-1\right )^{5}-1\right ) c_1
\]
✓ Mathematica. Time used: 3.854 (sec). Leaf size: 330
ode=6+12*x^2*y[x]^2+(7*x^3*y[x]+x/y[x])*D[y[x],x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \text {Root}\left [-\text {$\#$1}^7-\frac {3 \text {$\#$1}^5}{x^2}-\frac {3 \text {$\#$1}^3}{x^4}-\frac {\text {$\#$1}}{x^6}+\frac {e^{c_1}}{x^{12}}\&,1\right ]\\ y(x)&\to \text {Root}\left [-\text {$\#$1}^7-\frac {3 \text {$\#$1}^5}{x^2}-\frac {3 \text {$\#$1}^3}{x^4}-\frac {\text {$\#$1}}{x^6}+\frac {e^{c_1}}{x^{12}}\&,2\right ]\\ y(x)&\to \text {Root}\left [-\text {$\#$1}^7-\frac {3 \text {$\#$1}^5}{x^2}-\frac {3 \text {$\#$1}^3}{x^4}-\frac {\text {$\#$1}}{x^6}+\frac {e^{c_1}}{x^{12}}\&,3\right ]\\ y(x)&\to \text {Root}\left [-\text {$\#$1}^7-\frac {3 \text {$\#$1}^5}{x^2}-\frac {3 \text {$\#$1}^3}{x^4}-\frac {\text {$\#$1}}{x^6}+\frac {e^{c_1}}{x^{12}}\&,4\right ]\\ y(x)&\to \text {Root}\left [-\text {$\#$1}^7-\frac {3 \text {$\#$1}^5}{x^2}-\frac {3 \text {$\#$1}^3}{x^4}-\frac {\text {$\#$1}}{x^6}+\frac {e^{c_1}}{x^{12}}\&,5\right ]\\ y(x)&\to \text {Root}\left [-\text {$\#$1}^7-\frac {3 \text {$\#$1}^5}{x^2}-\frac {3 \text {$\#$1}^3}{x^4}-\frac {\text {$\#$1}}{x^6}+\frac {e^{c_1}}{x^{12}}\&,6\right ]\\ y(x)&\to \text {Root}\left [-\text {$\#$1}^7-\frac {3 \text {$\#$1}^5}{x^2}-\frac {3 \text {$\#$1}^3}{x^4}-\frac {\text {$\#$1}}{x^6}+\frac {e^{c_1}}{x^{12}}\&,7\right ] \end{align*}
✓ Sympy. Time used: 0.412 (sec). Leaf size: 29
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(12*x**2*y(x)**2 + (7*x**3*y(x) + x/y(x))*Derivative(y(x), x) + 6,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
- \log {\left (x \right )} - \frac {\log {\left (x y{\left (x \right )} \right )}}{5} - \frac {3 \log {\left (x^{2} y^{2}{\left (x \right )} + 1 \right )}}{5} = C_{1}
\]