82.2.4 problem Ex. 4
Internal
problem
ID
[18651]
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.
Exercises
at
page
14
Problem
number
:
Ex.
4
Date
solved
:
Thursday, March 13, 2025 at 12:27:48 PM
CAS
classification
:
[_separable]
\begin{align*} 3 \,{\mathrm e}^{x} \tan \left (y\right )+\left (1-{\mathrm e}^{x}\right ) \sec \left (y\right )^{2} y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.013 (sec). Leaf size: 205
ode:=3*exp(x)*tan(y(x))+(1-exp(x))*sec(y(x))^2*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\[
y \left (x \right ) = \frac {\arctan \left (-\frac {2 c_{1} \left ({\mathrm e}^{3 x}-3 \,{\mathrm e}^{2 x}+3 \,{\mathrm e}^{x}-1\right )}{-{\mathrm e}^{6 x} c_{1}^{2}+6 \,{\mathrm e}^{5 x} c_{1}^{2}-15 \,{\mathrm e}^{4 x} c_{1}^{2}+20 \,{\mathrm e}^{3 x} c_{1}^{2}-15 \,{\mathrm e}^{2 x} c_{1}^{2}+6 \,{\mathrm e}^{x} c_{1}^{2}-c_{1}^{2}-1}, \frac {{\mathrm e}^{6 x} c_{1}^{2}-6 \,{\mathrm e}^{5 x} c_{1}^{2}+15 \,{\mathrm e}^{4 x} c_{1}^{2}-20 \,{\mathrm e}^{3 x} c_{1}^{2}+15 \,{\mathrm e}^{2 x} c_{1}^{2}-6 \,{\mathrm e}^{x} c_{1}^{2}+c_{1}^{2}-1}{-{\mathrm e}^{6 x} c_{1}^{2}+6 \,{\mathrm e}^{5 x} c_{1}^{2}-15 \,{\mathrm e}^{4 x} c_{1}^{2}+20 \,{\mathrm e}^{3 x} c_{1}^{2}-15 \,{\mathrm e}^{2 x} c_{1}^{2}+6 \,{\mathrm e}^{x} c_{1}^{2}-c_{1}^{2}-1}\right )}{2}
\]
✓ Mathematica. Time used: 1.047 (sec). Leaf size: 74
ode=3*Exp[x]*Tan[y[x]]+(1-Exp[x])*Sec[y[x]]^2*D[y[x],x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to -\frac {1}{2} \arccos \left (-\tanh \left (3 \log \left (e^x-1\right )+2 c_1\right )\right ) \\
y(x)\to \frac {1}{2} \arccos \left (-\tanh \left (3 \log \left (e^x-1\right )+2 c_1\right )\right ) \\
y(x)\to 0 \\
y(x)\to -\frac {\pi }{2} \\
y(x)\to \frac {\pi }{2} \\
\end{align*}
✓ Sympy. Time used: 6.117 (sec). Leaf size: 223
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq((1 - exp(x))*Derivative(y(x), x)/cos(y(x))**2 + 3*exp(x)*tan(y(x)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = \pi - \frac {\operatorname {acos}{\left (\frac {e^{C_{1}} - 6 e^{C_{1} + x} + 15 e^{C_{1} + 2 x} - 20 e^{C_{1} + 3 x} + 15 e^{C_{1} + 4 x} - 6 e^{C_{1} + 5 x} + e^{C_{1} + 6 x} + 1}{- e^{C_{1}} + 6 e^{C_{1} + x} - 15 e^{C_{1} + 2 x} + 20 e^{C_{1} + 3 x} - 15 e^{C_{1} + 4 x} + 6 e^{C_{1} + 5 x} - e^{C_{1} + 6 x} + 1} \right )}}{2}, \ y{\left (x \right )} = \frac {\operatorname {acos}{\left (\frac {e^{C_{1}} - 6 e^{C_{1} + x} + 15 e^{C_{1} + 2 x} - 20 e^{C_{1} + 3 x} + 15 e^{C_{1} + 4 x} - 6 e^{C_{1} + 5 x} + e^{C_{1} + 6 x} + 1}{- e^{C_{1}} + 6 e^{C_{1} + x} - 15 e^{C_{1} + 2 x} + 20 e^{C_{1} + 3 x} - 15 e^{C_{1} + 4 x} + 6 e^{C_{1} + 5 x} - e^{C_{1} + 6 x} + 1} \right )}}{2}\right ]
\]