79.2.5 problem (f)

Internal problem ID [21087]
Book : Ordinary Differential Equations. By Wolfgang Walter. Graduate texts in Mathematics. Springer. NY. QA372.W224 1998
Section : Chapter 1. First order equations: Some integrable cases. Excercises XIII at page 24
Problem number : (f)
Date solved : Thursday, October 02, 2025 at 07:07:08 PM
CAS classification : [_separable]

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

With initial conditions

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

✓ Maple. Time used: 0.185 (sec). Leaf size: 23

ode:=diff(y(x),x) = cos(x)/cos(y(x))^2; 
ic:=[y(Pi) = 1/4*Pi]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \frac {\operatorname {RootOf}\left (2 \textit {\_Z} -\pi -2-8 \sin \left (x \right )+2 \sin \left (\textit {\_Z} \right )\right )}{2} \]

✓ Mathematica. Time used: 0.227 (sec). Leaf size: 36

ode=D[y[x],x]== Cos[x]/Cos[y[x]]^2; 
ic={y[Pi]==Pi/4}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \text {InverseFunction}\left [2 \left (\frac {\text {$\#$1}}{2}+\frac {1}{4} \sin (2 \text {$\#$1})\right )\&\right ]\left [\frac {1}{4} (8 \sin (x)+\pi +2)\right ] \end{align*}

✓ Sympy. Time used: 3.051 (sec). Leaf size: 26

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-cos(x)/cos(y(x))**2 + Derivative(y(x), x),0) 
ics = {y(pi): pi/4} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \frac {y{\left (x \right )}}{2} - \sin {\left (x \right )} + \frac {\sin {\left (y{\left (x \right )} \right )} \cos {\left (y{\left (x \right )} \right )}}{2} = \frac {1}{4} + \frac {\pi }{8} \]