74.4.23 problem 23

Internal problem ID [15916]
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 : Tuesday, January 28, 2025 at 08:20:31 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*}

✓ Solution by Maple

Time used: 0.059 (sec). Leaf size: 59

dsolve(tan(y(x))*sec(y(x))^2*diff(y(x),x)+cos(2*x)^3*sin(2*x)=0,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*}

✓ Solution by Mathematica

Time used: 2.910 (sec). Leaf size: 139

DSolve[Tan[y[x]]*Sec[y[x]]^2*D[y[x],x]+Cos[2*x]^3*Sin[2*x]==0,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*}