38.4.13 problem 4 (a)

Internal problem ID [8312]
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 : Tuesday, September 30, 2025 at 05:22:35 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*}

✓ Maple. Time used: 0.872 (sec). Leaf size: 72

ode:=diff(y(x),x) = sin(x)*cos(y(x)); 
ic:=[y(0) = 1]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.066 (sec). Leaf size: 55

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

\[ \text {Solve}\left [-y(x) \int _0^x0dK[1]+\int _0^x-\sec (y(x)) (\sin (K[1]-y(x))+\sin (K[1]+y(x)))dK[1]+2 \coth ^{-1}(\sin (y(x)))=2 \coth ^{-1}(\sin (1)),y(x)\right ] \]

✓ Sympy. Time used: 1.069 (sec). Leaf size: 97

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-sin(x)*cos(y(x)) + Derivative(y(x), x),0) 
ics = {y(0): 1} 
dsolve(ode,func=y(x),ics=ics)
 

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