87.5.8 problem 8

Internal problem ID [23301]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 1. Elementary methods. First order differential equations. Exercise at page 47
Problem number : 8
Date solved : Thursday, October 02, 2025 at 09:29:12 PM
CAS classification : [_separable]

\begin{align*} \cos \left (y\right )+\sin \left (x \right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.471 (sec). Leaf size: 65

ode:=cos(y(x))+sin(x)*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \arctan \left (\frac {-c_1^{2} \cos \left (x \right )+c_1^{2}-\cos \left (x \right )-1}{c_1^{2} \cos \left (x \right )-c_1^{2}-\cos \left (x \right )-1}, -\frac {2 c_1 \sin \left (x \right )}{c_1^{2} \cos \left (x \right )-c_1^{2}-\cos \left (x \right )-1}\right ) \]

✓ Mathematica. Time used: 0.14 (sec). Leaf size: 37

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

\begin{align*} y(x)&\to 2 \arctan \left (\tanh \left (\frac {1}{2} (\text {arctanh}(\cos (x))+c_1)\right )\right )\\ y(x)&\to -\frac {\pi }{2}\\ y(x)&\to \frac {\pi }{2} \end{align*}

✓ Sympy. Time used: 1.814 (sec). Leaf size: 56

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

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