28.1.6 problem 6

Internal problem ID [4312]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 2. First-Order and Simple Higher-Order Differential Equations. Page 78
Problem number : 6
Date solved : Tuesday, March 04, 2025 at 06:20:08 PM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 35

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

\begin{align*} y \left (x \right ) &= \operatorname {arccot}\left (\frac {1}{\sqrt {-x^{2}-2 c_{1}}}\right ) \\ y \left (x \right ) &= \frac {\pi }{2}+\arctan \left (\frac {1}{\sqrt {-x^{2}-2 c_{1}}}\right ) \\ \end{align*}

✓ Mathematica. Time used: 1.13 (sec). Leaf size: 103

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

\begin{align*} y(x)\to -\sec ^{-1}\left (-\sqrt {-x^2+8 c_1}\right ) \\ y(x)\to \sec ^{-1}\left (-\sqrt {-x^2+8 c_1}\right ) \\ y(x)\to -\sec ^{-1}\left (\sqrt {-x^2+8 c_1}\right ) \\ y(x)\to \sec ^{-1}\left (\sqrt {-x^2+8 c_1}\right ) \\ y(x)\to -\frac {\pi }{2} \\ y(x)\to \frac {\pi }{2} \\ \end{align*}

✓ Sympy. Time used: 2.187 (sec). Leaf size: 71

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

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