38.4.16 problem 4 (d)

Internal problem ID [8315]
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 (d)
Date solved : Tuesday, September 30, 2025 at 05:24:56 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 )&=-{\frac {5}{2}} \\ \end{align*}

✓ Maple. Time used: 0.655 (sec). Leaf size: 61

ode:=diff(y(x),x) = sin(x)*cos(y(x)); 
ic:=[y(0) = -5/2]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \arctan \left (\frac {\left (-1+\sin \left (\frac {5}{2}\right )\right ) {\mathrm e}^{2-2 \cos \left (x \right )}+\sin \left (\frac {5}{2}\right )+1}{\left (-1+\sin \left (\frac {5}{2}\right )\right ) {\mathrm e}^{2-2 \cos \left (x \right )}-\sin \left (\frac {5}{2}\right )-1}, \frac {\cos \left (\frac {5}{2}\right )}{\sin \left (\frac {5}{2}\right ) \sinh \left (-1+\cos \left (x \right )\right )+\cosh \left (-1+\cos \left (x \right )\right )}\right ) \]

✓ Mathematica. Time used: 0.031 (sec). Leaf size: 57

ode=D[y[x],x]==Sin[x]*Cos[y[x]]; 
ic={y[0]==-5/2}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [-y(x) \int _0^x0dK[1]+\int _0^x-\sec (y(x)) (\sin (K[1]-y(x))+\sin (K[1]+y(x)))dK[1]+2 \coth ^{-1}(\sin (y(x)))=-2 \coth ^{-1}\left (\sin \left (\frac {5}{2}\right )\right ),y(x)\right ] \]

✓ Sympy. Time used: 1.064 (sec). Leaf size: 110

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-sin(x)*cos(y(x)) + Derivative(y(x), x),0) 
ics = {y(0): -5/2} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = \pi - \operatorname {asin}{\left (\frac {e^{2 \cos {\left (x \right )}} + \frac {- e^{2} + e^{2} \sin {\left (\frac {5}{2} \right )}}{\sin {\left (\frac {5}{2} \right )} + 1}}{- e^{2 \cos {\left (x \right )}} + \frac {- e^{2} + e^{2} \sin {\left (\frac {5}{2} \right )}}{\sin {\left (\frac {5}{2} \right )} + 1}} \right )}, \ y{\left (x \right )} = \operatorname {asin}{\left (\frac {e^{2 \cos {\left (x \right )}} + \frac {- e^{2} + e^{2} \sin {\left (\frac {5}{2} \right )}}{\sin {\left (\frac {5}{2} \right )} + 1}}{- e^{2 \cos {\left (x \right )}} + \frac {- e^{2} + e^{2} \sin {\left (\frac {5}{2} \right )}}{\sin {\left (\frac {5}{2} \right )} + 1}} \right )}\right ] \]