56.1.1 problem 1

Internal problem ID [8713]
Book : Own collection of miscellaneous problems
Section : section 1.0
Problem number : 1
Date solved : Wednesday, March 05, 2025 at 06:13:16 AM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.096 (sec). Leaf size: 73

ode:=diff(y(x),x) = cos(y(x))*sec(x)/x; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \arctan \left (\frac {{\mathrm e}^{2 \left (\int \frac {\sec \left (x \right )}{x}d x \right )} c_{1}^{2}-1}{{\mathrm e}^{2 \left (\int \frac {\sec \left (x \right )}{x}d x \right )} c_{1}^{2}+1}, \frac {2 \,{\mathrm e}^{\int \frac {\sec \left (x \right )}{x}d x} c_{1}}{{\mathrm e}^{2 \left (\int \frac {\sec \left (x \right )}{x}d x \right )} c_{1}^{2}+1}\right ) \]

✓ Mathematica. Time used: 2.19 (sec). Leaf size: 49

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

\begin{align*} y(x)\to 2 \arctan \left (\tanh \left (\frac {1}{2} \left (\int _1^x\frac {\sec (K[1])}{K[1]}dK[1]+c_1\right )\right )\right ) \\ y(x)\to -\frac {\pi }{2} \\ y(x)\to \frac {\pi }{2} \\ \end{align*}

✓ Sympy. Time used: 63.016 (sec). Leaf size: 36

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

\[ \left [ y{\left (x \right )} = \pi - \operatorname {asin}{\left (\frac {1}{\tanh {\left (C_{1} + \int \frac {1}{x \cos {\left (x \right )}}\, dx \right )}} \right )}, \ y{\left (x \right )} = \operatorname {asin}{\left (\frac {1}{\tanh {\left (C_{1} + \int \frac {1}{x \cos {\left (x \right )}}\, dx \right )}} \right )}\right ] \]