85.33.31 problem 31

Internal problem ID [22654]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter two. First order and simple higher order ordinary differential equations. A Exercises at page 65
Problem number : 31
Date solved : Thursday, October 02, 2025 at 09:02:54 PM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.029 (sec). Leaf size: 9

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

\[ y = \arccos \left (c_1 \sec \left (x \right )\right ) \]

✓ Mathematica. Time used: 5.271 (sec). Leaf size: 47

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

\begin{align*} y(x)&\to -\arccos \left (\frac {1}{2} c_1 \sec (x)\right )\\ y(x)&\to \arccos \left (\frac {1}{2} c_1 \sec (x)\right )\\ y(x)&\to -\frac {\pi }{2}\\ y(x)&\to \frac {\pi }{2} \end{align*}

✓ Sympy. Time used: 0.357 (sec). Leaf size: 20

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

\[ \left [ y{\left (x \right )} = - \operatorname {acos}{\left (\frac {C_{1}}{\cos {\left (x \right )}} \right )} + 2 \pi , \ y{\left (x \right )} = \operatorname {acos}{\left (\frac {C_{1}}{\cos {\left (x \right )}} \right )}\right ] \]