44.4.13 problem 4 (a)

Internal problem ID [7026]
Book : A First Course in Differential Equations with Modeling Applications by Dennis G. Zill. 12 ed. Metric version. 2024. Cengage learning.
Section : Chapter 2. First order differential equations. Section 2.1 Solution curves without a solution. Exercises 2.1 at page 44
Problem number : 4 (a)
Date solved : Monday, January 27, 2025 at 02:41:19 PM
CAS classification : [_separable]

\begin{align*} y^{\prime }&=\sin \left (x \right ) \cos \left (y\right ) \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=1 \end{align*}

✓ Solution by Maple

Time used: 0.703 (sec). Leaf size: 88

dsolve([diff(y(x),x)=sin(x)*cos(y(x)),y(0) = 1],y(x), singsol=all)
 

\[ y = \arctan \left (\frac {\sin \left (1\right ) {\mathrm e}^{2-2 \cos \left (x \right )}+{\mathrm e}^{2-2 \cos \left (x \right )}+\sin \left (1\right )-1}{\sin \left (1\right ) {\mathrm e}^{2-2 \cos \left (x \right )}+{\mathrm e}^{2-2 \cos \left (x \right )}-\sin \left (1\right )+1}, \frac {2 \,{\mathrm e}^{1-\cos \left (x \right )} \cos \left (1\right )}{\sin \left (1\right ) {\mathrm e}^{2-2 \cos \left (x \right )}+{\mathrm e}^{2-2 \cos \left (x \right )}-\sin \left (1\right )+1}\right ) \]

✓ Solution by Mathematica

Time used: 0.727 (sec). Leaf size: 24

DSolve[{D[y[x],x]==Sin[x]*Cos[y[x]],{y[0]==1}},y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to 2 \arctan \left (\tanh \left (\text {arctanh}\left (\tan \left (\frac {1}{2}\right )\right )-\frac {\cos (x)}{2}+\frac {1}{2}\right )\right ) \]