81.3.13 problem 13
Internal
problem
ID
[18631]
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
:
Tuesday, January 28, 2025 at 12:04:51 PM
CAS
classification
:
[_Bernoulli]
\begin{align*} y^{\prime }-\tan \left (x \right ) y&=y^{4} \sec \left (x \right ) \end{align*}
✓ Solution by Maple
Time used: 0.019 (sec). Leaf size: 183
dsolve(diff(y(x),x)-tan(x)*y(x)=y(x)^4*sec(x),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*}
✓ Solution by Mathematica
Time used: 1.070 (sec). Leaf size: 109
DSolve[D[y[x],x]-Tan[x]*y[x]==y[x]^4*Sec[x],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*}