83.4.3 problem 1 (c)

Internal problem ID [21930]
Book : Differential Equations By Kaj L. Nielsen. Second edition 1966. Barnes and nobel. 66-28306
Section : Chapter III. First order differential equations of the first degree. Ex. V at page 42
Problem number : 1 (c)
Date solved : Thursday, October 02, 2025 at 08:14:18 PM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.373 (sec). Leaf size: 80

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

\[ y = \frac {\arctan \left (-\frac {2 c_1 \sin \left (2 x \right )}{c_1^{2} \cos \left (2 x \right )-c_1^{2}-\cos \left (2 x \right )-1}, \frac {c_1^{2} \cos \left (2 x \right )-c_1^{2}+\cos \left (2 x \right )+1}{c_1^{2} \cos \left (2 x \right )-c_1^{2}-\cos \left (2 x \right )-1}\right )}{2} \]

✓ Mathematica. Time used: 0.262 (sec). Leaf size: 68

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

\begin{align*} y(x)&\to -\frac {1}{2} \arccos (-\tanh (\text {arctanh}(\cos (2 x))+2 c_1))\\ y(x)&\to \frac {1}{2} \arccos (-\tanh (\text {arctanh}(\cos (2 x))+2 c_1))\\ y(x)&\to 0\\ y(x)&\to -\frac {\pi }{2}\\ y(x)&\to \frac {\pi }{2} \end{align*}

✓ Sympy. Time used: 5.625 (sec). Leaf size: 80

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

\[ \left [ y{\left (x \right )} = \pi - \frac {\operatorname {acos}{\left (\frac {- e^{4 C_{1}} \cos ^{2}{\left (x \right )} - \cos ^{2}{\left (x \right )} + 1}{e^{4 C_{1}} \cos ^{2}{\left (x \right )} - \cos ^{2}{\left (x \right )} + 1} \right )}}{2}, \ y{\left (x \right )} = \frac {\operatorname {acos}{\left (\frac {e^{4 C_{1}} \cos ^{2}{\left (x \right )} + \cos ^{2}{\left (x \right )} - 1}{- e^{4 C_{1}} \cos ^{2}{\left (x \right )} + \cos ^{2}{\left (x \right )} - 1} \right )}}{2}\right ] \]