19.1.8 problem 8

Internal problem ID [3522]
Book : Differential equations and linear algebra, Stephen W. Goode, second edition, 2000
Section : 1.4, page 36
Problem number : 8
Date solved : Sunday, March 30, 2025 at 01:45:48 AM
CAS classification : [_separable]

\begin{align*} y^{\prime }&=\frac {\cos \left (x -y\right )}{\sin \left (x \right ) \sin \left (y\right )}-1 \end{align*}

✓ Maple. Time used: 0.040 (sec). Leaf size: 11

ode:=diff(y(x),x) = cos(x-y(x))/sin(x)/sin(y(x))-1; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \arccos \left (\frac {\csc \left (x \right )}{c_1}\right ) \]

✓ Mathematica. Time used: 5.88 (sec). Leaf size: 47

ode=D[y[x],x]==(Cos[x-y[x]])/(Sin[x]*Sin[y[x]])-1; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.592 (sec). Leaf size: 20

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), x) + 1 - cos(x - y(x))/(sin(x)*sin(y(x))),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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