82.12.4 problem Ex. 4

Internal problem ID [18698]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter II. Equations of the first order and of the first degree. Examples on chapter II at page 29
Problem number : Ex. 4
Date solved : Monday, March 31, 2025 at 05:58:33 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: 0.462 (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: 0.419 (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: 8.217 (sec). Leaf size: 80

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

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