50.4.6 problem 6

Internal problem ID [7855]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 1. What is a differential equation. Section 1.5. Exact Equations. Page 20
Problem number : 6
Date solved : Sunday, March 30, 2025 at 12:30:49 PM
CAS classification : [_separable]

\begin{align*} \cos \left (x \right ) \cos \left (y\right )^{2}+2 \sin \left (x \right ) \sin \left (y\right ) \cos \left (y\right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.098 (sec). Leaf size: 31

ode:=cos(x)*cos(y(x))^2+2*sin(x)*sin(y(x))*cos(y(x))*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {\pi }{2} \\ y &= \arccos \left (\sqrt {\sin \left (x \right ) c_1}\right ) \\ y &= \frac {\pi }{2}+\arcsin \left (\sqrt {\sin \left (x \right ) c_1}\right ) \\ \end{align*}

✓ Mathematica. Time used: 5.211 (sec). Leaf size: 73

ode=(Cos[x]*Cos[y[x]]^2)+(2*Sin[x]*Sin[y[x]]*Cos[y[x]])*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to -\frac {\pi }{2} \\ y(x)\to \frac {\pi }{2} \\ y(x)\to -\arccos \left (-\frac {1}{4} c_1 \sqrt {\sin (x)}\right ) \\ y(x)\to \arccos \left (-\frac {1}{4} c_1 \sqrt {\sin (x)}\right ) \\ y(x)\to -\frac {\pi }{2} \\ y(x)\to \frac {\pi }{2} \\ \end{align*}

✓ Sympy. Time used: 0.839 (sec). Leaf size: 27

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

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