23.1.96 problem 94
Internal
problem
ID
[4703]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Part
II.
Chapter
1.
THE
DIFFERENTIAL
EQUATION
IS
OF
FIRST
ORDER
AND
OF
FIRST
DEGREE,
page
223
Problem
number
:
94
Date
solved
:
Tuesday, September 30, 2025 at 08:15:37 AM
CAS
classification
:
[_Bernoulli]
\begin{align*} y^{\prime }&=\left (\tan \left (x \right )+y^{3} \sec \left (x \right )\right ) y \end{align*}
✓ Maple. Time used: 0.012 (sec). Leaf size: 185
ode:=diff(y(x),x) = (tan(x)+y(x)^3*sec(x))*y(x);
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {\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}} \sec \left (x \right )}{\cos \left (x \right )^{3} c_1 -2 \cos \left (x \right )^{2} \sin \left (x \right )-\sin \left (x \right )} \\
y &= -\frac {\left (1+i \sqrt {3}\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}} \sec \left (x \right )}{2 \cos \left (x \right )^{3} c_1 -4 \cos \left (x \right )^{2} \sin \left (x \right )-2 \sin \left (x \right )} \\
y &= \frac {\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 (-1+i \sqrt {3}\right ) \sec \left (x \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: 0.945 (sec). Leaf size: 238
ode=D[y[x],x]==(Tan[x]+y[x]^3+Sec[x])*y[x];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to -\frac {\sqrt [3]{-5} 2^{2/3}}{\sqrt [3]{-30 \sin (x)+12 \sin (2 x)+2 \sin (3 x)-30 \cos (x)+2 \cos (3 x)-6 (2+5 c_1) \cos (2 x)-75 c_1 \sin (x)+5 c_1 \sin (3 x)+20+50 c_1}}\\ y(x)&\to \frac {2^{2/3} \sqrt [3]{5}}{\sqrt [3]{-30 \sin (x)+12 \sin (2 x)+2 \sin (3 x)-30 \cos (x)+2 \cos (3 x)-6 (2+5 c_1) \cos (2 x)-75 c_1 \sin (x)+5 c_1 \sin (3 x)+20+50 c_1}}\\ y(x)&\to \frac {(-2)^{2/3} \sqrt [3]{5}}{\sqrt [3]{-30 \sin (x)+12 \sin (2 x)+2 \sin (3 x)-30 \cos (x)+2 \cos (3 x)-6 (2+5 c_1) \cos (2 x)-75 c_1 \sin (x)+5 c_1 \sin (3 x)+20+50 c_1}}\\ y(x)&\to 0 \end{align*}
✓ Sympy. Time used: 5.471 (sec). Leaf size: 104
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-(y(x)**3*sec(x) + tan(x))*y(x) + Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = \frac {\left (-1 - \sqrt {3} i\right ) \sqrt [3]{\frac {1}{C_{1} \cos ^{3}{\left (x \right )} - 2 \sin {\left (x \right )} \cos ^{2}{\left (x \right )} - \sin {\left (x \right )}}}}{2}, \ y{\left (x \right )} = \frac {\left (-1 + \sqrt {3} i\right ) \sqrt [3]{\frac {1}{C_{1} \cos ^{3}{\left (x \right )} - 2 \sin {\left (x \right )} \cos ^{2}{\left (x \right )} - \sin {\left (x \right )}}}}{2}, \ y{\left (x \right )} = \sqrt [3]{\frac {1}{C_{1} \cos ^{3}{\left (x \right )} - 2 \sin {\left (x \right )} \cos ^{2}{\left (x \right )} - \sin {\left (x \right )}}}\right ]
\]