74.4.23 problem 23
Internal
problem
ID
[15837]
Book
:
INTRODUCTORY
DIFFERENTIAL
EQUATIONS.
Martha
L.
Abell,
James
P.
Braselton.
Fourth
edition
2014.
ElScAe.
2014
Section
:
Chapter
2.
First
Order
Equations.
Exercises
2.2,
page
39
Problem
number
:
23
Date
solved
:
Thursday, March 13, 2025 at 06:48:47 AM
CAS
classification
:
[_separable]
\begin{align*} \tan \left (y\right ) \sec \left (y\right )^{2} y^{\prime }+\cos \left (2 x \right )^{3} \sin \left (2 x \right )&=0 \end{align*}
✓ Maple. Time used: 0.059 (sec). Leaf size: 59
ode:=tan(y(x))*sec(y(x))^2*diff(y(x),x)+cos(2*x)^3*sin(2*x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \operatorname {arccot}\left (\frac {8}{\sqrt {-2+4 \cos \left (4 x \right )^{2}-256 c_{1} +8 \cos \left (4 x \right )}}\right ) \\
y &= \frac {\pi }{2}+\arctan \left (\frac {8}{\sqrt {-2+4 \cos \left (4 x \right )^{2}-256 c_{1} +8 \cos \left (4 x \right )}}\right ) \\
\end{align*}
✓ Mathematica. Time used: 2.91 (sec). Leaf size: 139
ode=Tan[y[x]]*Sec[y[x]]^2*D[y[x],x]+Cos[2*x]^3*Sin[2*x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to -\sec ^{-1}\left (-\frac {\sqrt {8 \cos ^4(2 x)+c_1}}{4 \sqrt {2}}\right ) \\
y(x)\to \sec ^{-1}\left (-\frac {\sqrt {8 \cos ^4(2 x)+c_1}}{4 \sqrt {2}}\right ) \\
y(x)\to -\sec ^{-1}\left (\frac {\sqrt {8 \cos ^4(2 x)+c_1}}{4 \sqrt {2}}\right ) \\
y(x)\to \sec ^{-1}\left (\frac {\sqrt {8 \cos ^4(2 x)+c_1}}{4 \sqrt {2}}\right ) \\
y(x)\to -\frac {\pi }{2} \\
y(x)\to \frac {\pi }{2} \\
\end{align*}
✓ Sympy. Time used: 14.494 (sec). Leaf size: 85
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(sin(2*x)*cos(2*x)**3 + tan(y(x))*Derivative(y(x), x)/cos(y(x))**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = - \operatorname {acos}{\left (- 2 \sqrt {\frac {1}{C_{1} + \cos ^{4}{\left (2 x \right )}}} \right )} + 2 \pi , \ y{\left (x \right )} = - \operatorname {acos}{\left (2 \sqrt {\frac {1}{C_{1} + \cos ^{4}{\left (2 x \right )}}} \right )} + 2 \pi , \ y{\left (x \right )} = \operatorname {acos}{\left (- 2 \sqrt {\frac {1}{C_{1} + \cos ^{4}{\left (2 x \right )}}} \right )}, \ y{\left (x \right )} = \operatorname {acos}{\left (2 \sqrt {\frac {1}{C_{1} + \cos ^{4}{\left (2 x \right )}}} \right )}\right ]
\]