80.3.4 problem 4

Internal problem ID [18469]
Book : Elementary Differential Equations. By Thornton C. Fry. D Van Nostrand. NY. First Edition (1929)
Section : Chapter IV. Methods of solution: First order equations. section 29. Problems at page 81
Problem number : 4
Date solved : Thursday, March 13, 2025 at 12:03:07 PM
CAS classification : [_separable]

\begin{align*} \sec \left (x \right )^{2} \tan \left (y\right ) y^{\prime }+\sec \left (y\right )^{2} \tan \left (x \right )&=0 \end{align*}

✓ Maple. Time used: 0.010 (sec). Leaf size: 41

ode:=sec(x)^2*tan(y(x))*diff(y(x),x)+sec(y(x))^2*tan(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y \left (x \right ) &= \operatorname {arcsec}\left (\frac {2}{\sqrt {-2 \cos \left (2 x \right )+8 c_{1}}}\right ) \\ y \left (x \right ) &= \frac {\pi }{2}+\operatorname {arccsc}\left (\frac {2}{\sqrt {-2 \cos \left (2 x \right )+8 c_{1}}}\right ) \\ \end{align*}

✓ Mathematica. Time used: 0.504 (sec). Leaf size: 41

ode=Sec[x]^2*Tan[y[x]]*D[y[x],x]+Sec[y[x]]^2*Tan[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to -\frac {1}{2} \arccos (-\cos (2 x)-2 c_1) \\ y(x)\to \frac {1}{2} \arccos (-\cos (2 x)-2 c_1) \\ \end{align*}

✓ Sympy. Time used: 0.635 (sec). Leaf size: 26

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

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