19.1.14 problem 14

Internal problem ID [3528]
Book : Differential equations and linear algebra, Stephen W. Goode, second edition, 2000
Section : 1.4, page 36
Problem number : 14
Date solved : Sunday, March 30, 2025 at 01:46:07 AM
CAS classification : [_separable]

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

With initial conditions

\begin{align*} y \left (\frac {\pi }{4}\right )&=\frac {\pi }{4} \end{align*}

✓ Maple. Time used: 0.303 (sec). Leaf size: 9

ode:=diff(y(x),x) = 1-sin(x+y(x))/sin(y(x))/cos(x); 
ic:=y(1/4*Pi) = 1/4*Pi; 
dsolve([ode,ic],y(x), singsol=all);
 

\[ y = \arccos \left (\frac {\sec \left (x \right )}{2}\right ) \]

✓ Mathematica. Time used: 5.987 (sec). Leaf size: 12

ode=D[y[x],x]==1- Sin[x+y[x]]/(Sin[y[x]]*Cos[x]); 
ic=y[Pi/4]==Pi/4; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \arccos \left (\frac {\sec (x)}{2}\right ) \]

✓ Sympy. Time used: 0.577 (sec). Leaf size: 10

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(sin(x + y(x))/(sin(y(x))*cos(x)) + Derivative(y(x), x) - 1,0) 
ics = {y(pi/4): pi/4} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \operatorname {acos}{\left (\frac {1}{2 \cos {\left (x \right )}} \right )} \]