28.1.3 problem 3
Internal
problem
ID
[4309]
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
:
3
Date
solved
:
Tuesday, March 04, 2025 at 06:19:53 PM
CAS
classification
:
[_separable]
\begin{align*} x \cos \left (y\right )^{2}+{\mathrm e}^{x} \tan \left (y\right ) y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.012 (sec). Leaf size: 77
ode:=x*cos(y(x))^2+exp(x)*tan(y(x))*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y \left (x \right ) &= \pi -\operatorname {arccot}\left (\frac {\sqrt {2}\, \sqrt {{\mathrm e}^{x} \left (x +1-{\mathrm e}^{x} c_{1} \right )}}{2 x +2-2 \,{\mathrm e}^{x} c_{1}}\right ) \\
y \left (x \right ) &= \frac {\pi }{2}-\arctan \left (\frac {\sqrt {2}\, \sqrt {{\mathrm e}^{x} \left (x +1-{\mathrm e}^{x} c_{1} \right )}}{2 x +2-2 \,{\mathrm e}^{x} c_{1}}\right ) \\
\end{align*}
✓ Mathematica. Time used: 15.004 (sec). Leaf size: 149
ode=x*Cos[y[x]]^2+Exp[x]*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 {2} \sqrt {e^{-x} \left (x+4 c_1 e^x+1\right )}\right ) \\
y(x)\to \sec ^{-1}\left (-\sqrt {2} \sqrt {e^{-x} \left (x+4 c_1 e^x+1\right )}\right ) \\
y(x)\to -\sec ^{-1}\left (\sqrt {2} \sqrt {e^{-x} \left (x+4 c_1 e^x+1\right )}\right ) \\
y(x)\to \sec ^{-1}\left (\sqrt {2} \sqrt {e^{-x} \left (x+4 c_1 e^x+1\right )}\right ) \\
y(x)\to -\frac {\pi }{2} \\
y(x)\to \frac {\pi }{2} \\
\end{align*}
✓ Sympy. Time used: 4.191 (sec). Leaf size: 112
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x*cos(y(x))**2 + exp(x)*tan(y(x))*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = - \operatorname {acos}{\left (- \frac {\sqrt {2} \sqrt {\frac {e^{x}}{C_{1} e^{x} + x + 1}}}{2} \right )} + 2 \pi , \ y{\left (x \right )} = - \operatorname {acos}{\left (\frac {\sqrt {2} \sqrt {\frac {e^{x}}{C_{1} e^{x} + x + 1}}}{2} \right )} + 2 \pi , \ y{\left (x \right )} = \operatorname {acos}{\left (- \frac {\sqrt {2} \sqrt {\frac {e^{x}}{C_{1} e^{x} + x + 1}}}{2} \right )}, \ y{\left (x \right )} = \operatorname {acos}{\left (\frac {\sqrt {2} \sqrt {\frac {e^{x}}{C_{1} e^{x} + x + 1}}}{2} \right )}\right ]
\]