68.4.23 problem 23

Internal problem ID [17203]
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, October 02, 2025 at 01:52:25 PM
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: 0.009 (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: 1.81 (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: 9.342 (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)
 

\[ \left [ y{\left (x \right )} = - \operatorname {acos}{\left (- 2 \sqrt {\frac {1}{C_{1} + \cos ^{4}{\left (2 x \right )}}} \right )} + 2 \pi , \ y{\left (x \right )} = - \operatorname {acos}{\left (2 \sqrt {\frac {1}{C_{1} + \cos ^{4}{\left (2 x \right )}}} \right )} + 2 \pi , \ y{\left (x \right )} = \operatorname {acos}{\left (- 2 \sqrt {\frac {1}{C_{1} + \cos ^{4}{\left (2 x \right )}}} \right )}, \ y{\left (x \right )} = \operatorname {acos}{\left (2 \sqrt {\frac {1}{C_{1} + \cos ^{4}{\left (2 x \right )}}} \right )}\right ] \]