81.3.13 problem 13
Internal
problem
ID
[18552]
Book
:
A
short
course
on
differential
equations.
By
Donald
Francis
Campbell.
Maxmillan
company.
London.
1907
Section
:
Chapter
III.
Ordinary
differential
equations
of
the
first
order
and
first
degree.
Exercises
at
page
33
Problem
number
:
13
Date
solved
:
Thursday, March 13, 2025 at 12:12:48 PM
CAS
classification
:
[_Bernoulli]
\begin{align*} y^{\prime }-\tan \left (x \right ) y&=y^{4} \sec \left (x \right ) \end{align*}
✓ Maple. Time used: 0.019 (sec). Leaf size: 183
ode:=diff(y(x),x)-tan(x)*y(x) = y(x)^4*sec(x);
dsolve(ode,y(x), singsol=all);
\begin{align*}
y \left (x \right ) &= \frac {\sec \left (x \right ) \left (\cos \left (x \right )^{3} \left (\cos \left (x \right )^{3} c_{1} -2 \cos \left (x \right )^{2} \sin \left (x \right )-\sin \left (x \right )\right )^{2}\right )^{{1}/{3}}}{\cos \left (x \right )^{3} c_{1} -2 \cos \left (x \right )^{2} \sin \left (x \right )-\sin \left (x \right )} \\
y \left (x \right ) &= -\frac {\left (1+i \sqrt {3}\right ) \sec \left (x \right ) \left (\cos \left (x \right )^{3} \left (\cos \left (x \right )^{3} c_{1} -2 \cos \left (x \right )^{2} \sin \left (x \right )-\sin \left (x \right )\right )^{2}\right )^{{1}/{3}}}{2 \cos \left (x \right )^{3} c_{1} -4 \cos \left (x \right )^{2} \sin \left (x \right )-2 \sin \left (x \right )} \\
y \left (x \right ) &= \frac {\sec \left (x \right ) \left (\cos \left (x \right )^{3} \left (\cos \left (x \right )^{3} c_{1} -2 \cos \left (x \right )^{2} \sin \left (x \right )-\sin \left (x \right )\right )^{2}\right )^{{1}/{3}} \left (i \sqrt {3}-1\right )}{2 \cos \left (x \right )^{3} c_{1} -4 \cos \left (x \right )^{2} \sin \left (x \right )-2 \sin \left (x \right )} \\
\end{align*}
✓ Mathematica. Time used: 1.07 (sec). Leaf size: 109
ode=D[y[x],x]-Tan[x]*y[x]==y[x]^4*Sec[x];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \frac {1}{\sqrt [3]{-\sin ^3(x)+c_1 \cos ^3(x)-3 \sin (x) \cos ^2(x)}} \\
y(x)\to -\frac {\sqrt [3]{-1}}{\sqrt [3]{-\sin ^3(x)+c_1 \cos ^3(x)-3 \sin (x) \cos ^2(x)}} \\
y(x)\to \frac {(-1)^{2/3}}{\sqrt [3]{-\sin ^3(x)+c_1 \cos ^3(x)-3 \sin (x) \cos ^2(x)}} \\
y(x)\to 0 \\
\end{align*}
✓ Sympy. Time used: 7.964 (sec). Leaf size: 104
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-y(x)**4/cos(x) - y(x)*tan(x) + Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = \frac {\left (-1 - \sqrt {3} i\right ) \sqrt [3]{\frac {1}{C_{1} \cos ^{3}{\left (x \right )} - 2 \sin {\left (x \right )} \cos ^{2}{\left (x \right )} - \sin {\left (x \right )}}}}{2}, \ y{\left (x \right )} = \frac {\left (-1 + \sqrt {3} i\right ) \sqrt [3]{\frac {1}{C_{1} \cos ^{3}{\left (x \right )} - 2 \sin {\left (x \right )} \cos ^{2}{\left (x \right )} - \sin {\left (x \right )}}}}{2}, \ y{\left (x \right )} = \sqrt [3]{\frac {1}{C_{1} \cos ^{3}{\left (x \right )} - 2 \sin {\left (x \right )} \cos ^{2}{\left (x \right )} - \sin {\left (x \right )}}}\right ]
\]