43.22.4 problem 1(d)

Internal problem ID [9036]
Book : An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY 1961
Section : Chapter 5. Existence and uniqueness of solutions to first order equations. Page 198
Problem number : 1(d)
Date solved : Tuesday, September 30, 2025 at 06:02:18 PM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.083 (sec). Leaf size: 25

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

\begin{align*} y &= \arccos \left (\frac {1}{\sqrt {c_1 \sin \left (x \right )}}\right ) \\ y &= \frac {\pi }{2}+\arcsin \left (\frac {1}{\sqrt {c_1 \sin \left (x \right )}}\right ) \\ \end{align*}

✓ Mathematica. Time used: 5.369 (sec). Leaf size: 73

ode=Cos[x]*Cos[y[x]]^2-Sin[x]*Sin[2*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 {c_1}{4 \sqrt {\sin (x)}}\right )\\ y(x)&\to \arccos \left (-\frac {c_1}{4 \sqrt {\sin (x)}}\right )\\ y(x)&\to -\frac {\pi }{2}\\ y(x)&\to \frac {\pi }{2} \end{align*}

✓ Sympy. Time used: 0.380 (sec). Leaf size: 27

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

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