86.1.7 problem 7

Internal problem ID [23069]
Book : An introduction to Differential Equations. By Howard Frederick Cleaves. 1969. Oliver and Boyd publisher. ISBN 0050015044
Section : Chapter 3. Some standard types of differential equations. Exercise 3b at page 43
Problem number : 7
Date solved : Thursday, October 02, 2025 at 09:18:41 PM
CAS classification : [_separable]

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

With initial conditions

\begin{align*} y \left (\frac {\pi }{4}\right )&=\frac {\pi }{2} \\ \end{align*}

✓ Maple. Time used: 0.177 (sec). Leaf size: 23

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

\begin{align*} y &= \arcsin \left ({\mathrm e}^{\tan \left (x \right )-1}\right ) \\ y &= -\arcsin \left ({\mathrm e}^{\tan \left (x \right )-1}\right )+\pi \\ \end{align*}

✓ Mathematica. Time used: 34.205 (sec). Leaf size: 12

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

\begin{align*} y(x)&\to \arcsin \left (e^{\tan (x)-1}\right ) \end{align*}

✓ Sympy. Time used: 0.494 (sec). Leaf size: 26

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

\[ \left [ y{\left (x \right )} = \pi - \operatorname {asin}{\left (\frac {e^{\tan {\left (x \right )}}}{e} \right )}, \ y{\left (x \right )} = \operatorname {asin}{\left (\frac {e^{\tan {\left (x \right )}}}{e} \right )}\right ] \]