63.1.131 problem 190
Internal
problem
ID
[15571]
Book
:
DIFFERENTIAL
and
INTEGRAL
CALCULUS.
VOL
I.
by
N.
PISKUNOV.
MIR
PUBLISHERS,
Moscow
1969.
Section
:
Chapter
8.
Differential
equations.
Exercises
page
595
Problem
number
:
190
Date
solved
:
Thursday, October 02, 2025 at 10:20:14 AM
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: 0.020 (sec). Leaf size: 192
ode:=3*exp(x)*tan(y(x))+(-exp(x)+1)*sec(y(x))^2*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\[
y = \frac {\arctan \left (-\frac {2 c_1 \left (-1+{\mathrm e}^{x}\right )^{3}}{{\mathrm e}^{6 x} c_1^{2}-6 \,{\mathrm e}^{5 x} c_1^{2}+15 \,{\mathrm e}^{4 x} c_1^{2}-20 \,{\mathrm e}^{3 x} c_1^{2}+15 \,{\mathrm e}^{2 x} c_1^{2}-6 \,{\mathrm e}^{x} c_1^{2}+c_1^{2}+1}, \frac {{\mathrm e}^{6 x} c_1^{2}-6 \,{\mathrm e}^{5 x} c_1^{2}+15 \,{\mathrm e}^{4 x} c_1^{2}-20 \,{\mathrm e}^{3 x} c_1^{2}+15 \,{\mathrm e}^{2 x} c_1^{2}-6 \,{\mathrm e}^{x} c_1^{2}+c_1^{2}-1}{-{\mathrm e}^{6 x} c_1^{2}+6 \,{\mathrm e}^{5 x} c_1^{2}-15 \,{\mathrm e}^{4 x} c_1^{2}+20 \,{\mathrm e}^{3 x} c_1^{2}-15 \,{\mathrm e}^{2 x} c_1^{2}+6 \,{\mathrm e}^{x} c_1^{2}-c_1^{2}-1}\right )}{2}
\]
✓ Mathematica. Time used: 0.622 (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: 3.662 (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)
\[
\left [ y{\left (x \right )} = \pi - \frac {\operatorname {acos}{\left (\frac {e^{C_{1}} - 6 e^{C_{1} + x} + 15 e^{C_{1} + 2 x} - 20 e^{C_{1} + 3 x} + 15 e^{C_{1} + 4 x} - 6 e^{C_{1} + 5 x} + e^{C_{1} + 6 x} + 1}{- e^{C_{1}} + 6 e^{C_{1} + x} - 15 e^{C_{1} + 2 x} + 20 e^{C_{1} + 3 x} - 15 e^{C_{1} + 4 x} + 6 e^{C_{1} + 5 x} - e^{C_{1} + 6 x} + 1} \right )}}{2}, \ y{\left (x \right )} = \frac {\operatorname {acos}{\left (\frac {e^{C_{1}} - 6 e^{C_{1} + x} + 15 e^{C_{1} + 2 x} - 20 e^{C_{1} + 3 x} + 15 e^{C_{1} + 4 x} - 6 e^{C_{1} + 5 x} + e^{C_{1} + 6 x} + 1}{- e^{C_{1}} + 6 e^{C_{1} + x} - 15 e^{C_{1} + 2 x} + 20 e^{C_{1} + 3 x} - 15 e^{C_{1} + 4 x} + 6 e^{C_{1} + 5 x} - e^{C_{1} + 6 x} + 1} \right )}}{2}\right ]
\]