15.8.30 problem 31
Internal
problem
ID
[3033]
Book
:
Differential
Equations
by
Alfred
L.
Nelson,
Karl
W.
Folley,
Max
Coral.
3rd
ed.
DC
heath.
Boston.
1964
Section
:
Exercise
12,
page
46
Problem
number
:
31
Date
solved
:
Tuesday, March 04, 2025 at 03:46:12 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 } \end{align*}
✓ Maple. Time used: 0.053 (sec). Leaf size: 144
ode:=3*exp(x)*tan(y(x)) = (1-exp(x))*sec(y(x))^2*diff(y(x),x);
dsolve(ode,y(x), singsol=all);
\[
y = \frac {\arctan \left (\frac {2 c_{1} \left ({\mathrm e}^{3 x}-3 \,{\mathrm e}^{2 x}+3 \,{\mathrm e}^{x}-1\right )}{{\mathrm e}^{6 x}-6 \,{\mathrm e}^{5 x}+15 \,{\mathrm e}^{4 x}-20 \,{\mathrm e}^{3 x}+15 \,{\mathrm e}^{2 x}+c_{1}^{2}-6 \,{\mathrm e}^{x}+1}, \frac {{\mathrm e}^{6 x}-6 \,{\mathrm e}^{5 x}+15 \,{\mathrm e}^{4 x}-20 \,{\mathrm e}^{3 x}+15 \,{\mathrm e}^{2 x}-c_{1}^{2}-6 \,{\mathrm e}^{x}+1}{{\mathrm e}^{6 x}-6 \,{\mathrm e}^{5 x}+15 \,{\mathrm e}^{4 x}-20 \,{\mathrm e}^{3 x}+15 \,{\mathrm e}^{2 x}+c_{1}^{2}-6 \,{\mathrm e}^{x}+1}\right )}{2}
\]
✓ Mathematica. Time used: 1.295 (sec). Leaf size: 78
ode=3*Exp[x]*Tan[y[x]]==(1-Exp[x])*Sec[y[x]]^2*D[y[x],x];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to -\frac {1}{2} \arccos \left (-\tanh \left (-3 \log \left (2-2 e^x\right )+2 c_1\right )\right ) \\
y(x)\to \frac {1}{2} \arccos \left (-\tanh \left (-3 \log \left (2-2 e^x\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: 2.789 (sec). Leaf size: 175
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq((exp(x) - 1)*Derivative(y(x), x)/cos(y(x))**2 + 3*exp(x)*tan(y(x)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = \pi - \frac {\operatorname {acos}{\left (\frac {- C_{1} - e^{6 x} + 6 e^{5 x} - 15 e^{4 x} + 20 e^{3 x} - 15 e^{2 x} + 6 e^{x} - 1}{C_{1} - e^{6 x} + 6 e^{5 x} - 15 e^{4 x} + 20 e^{3 x} - 15 e^{2 x} + 6 e^{x} - 1} \right )}}{2}, \ y{\left (x \right )} = \frac {\operatorname {acos}{\left (\frac {- C_{1} - e^{6 x} + 6 e^{5 x} - 15 e^{4 x} + 20 e^{3 x} - 15 e^{2 x} + 6 e^{x} - 1}{C_{1} - e^{6 x} + 6 e^{5 x} - 15 e^{4 x} + 20 e^{3 x} - 15 e^{2 x} + 6 e^{x} - 1} \right )}}{2}\right ]
\]