83.3.12 problem 12
Internal
problem
ID
[18989]
Book
:
A
Text
book
for
differentional
equations
for
postgraduate
students
by
Ray
and
Chaturvedi.
First
edition,
1958.
BHASKAR
press.
INDIA
Section
:
Chapter
II.
Equations
of
first
order
and
first
degree.
Exercise
II
(B)
at
page
9
Problem
number
:
12
Date
solved
:
Thursday, March 13, 2025 at 01:17:40 PM
CAS
classification
:
[_separable]
\begin{align*} \cos \left (y\right ) \ln \left (\sec \left (x \right )+\tan \left (x \right )\right )&=\cos \left (x \right ) \ln \left (\sec \left (y\right )+\tan \left (y\right )\right ) y^{\prime } \end{align*}
✓ Maple. Time used: 0.323 (sec). Leaf size: 209
ode:=cos(y(x))*ln(sec(x)+tan(x)) = cos(x)*ln(sec(y(x))+tan(y(x)))*diff(y(x),x);
dsolve(ode,y(x), singsol=all);
\begin{align*}
y \left (x \right ) &= \arctan \left (\frac {{\mathrm e}^{2 \sqrt {\ln \left (-\frac {\cos \left (x \right )}{-1+\sin \left (x \right )}\right )^{2}+2 c_{1}}}-1}{{\mathrm e}^{2 \sqrt {\ln \left (-\frac {\cos \left (x \right )}{-1+\sin \left (x \right )}\right )^{2}+2 c_{1}}}+1}, \frac {2 \,{\mathrm e}^{\sqrt {\ln \left (-\frac {\cos \left (x \right )}{-1+\sin \left (x \right )}\right )^{2}+2 c_{1}}}}{{\mathrm e}^{2 \sqrt {\ln \left (-\frac {\cos \left (x \right )}{-1+\sin \left (x \right )}\right )^{2}+2 c_{1}}}+1}\right ) \\
y \left (x \right ) &= \arctan \left (\frac {{\mathrm e}^{-2 \sqrt {\ln \left (-\frac {\cos \left (x \right )}{-1+\sin \left (x \right )}\right )^{2}+2 c_{1}}}-1}{{\mathrm e}^{-2 \sqrt {\ln \left (-\frac {\cos \left (x \right )}{-1+\sin \left (x \right )}\right )^{2}+2 c_{1}}}+1}, \frac {2 \,{\mathrm e}^{-\sqrt {\ln \left (-\frac {\cos \left (x \right )}{-1+\sin \left (x \right )}\right )^{2}+2 c_{1}}}}{{\mathrm e}^{-2 \sqrt {\ln \left (-\frac {\cos \left (x \right )}{-1+\sin \left (x \right )}\right )^{2}+2 c_{1}}}+1}\right ) \\
\end{align*}
✓ Mathematica. Time used: 60.159 (sec). Leaf size: 103
ode=Cos[y[x]]*Log[ Sec[x]+Tan[x] ]==Cos[x]*Log[Sec[y[x]]+Tan[y[x]]]*D[y[x],x];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to -\sec ^{-1}\left (\frac {1}{2} \left (e^{-\sqrt {\log ^2(\tan (x)+\sec (x))+2 c_1}}+e^{\sqrt {\log ^2(\tan (x)+\sec (x))+2 c_1}}\right )\right ) \\
y(x)\to \sec ^{-1}\left (\frac {1}{2} \left (e^{-\sqrt {\log ^2(\tan (x)+\sec (x))+2 c_1}}+e^{\sqrt {\log ^2(\tan (x)+\sec (x))+2 c_1}}\right )\right ) \\
\end{align*}
✓ Sympy. Time used: 55.131 (sec). Leaf size: 32
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(log(tan(x) + 1/cos(x))*cos(y(x)) - log(tan(y(x)) + 1/cos(y(x)))*cos(x)*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\int \limits ^{y{\left (x \right )}} \frac {\log {\left (\tan {\left (y \right )} + \frac {1}{\cos {\left (y \right )}} \right )}}{\cos {\left (y \right )}}\, dy = C_{1} + \int \frac {\log {\left (\tan {\left (x \right )} + \frac {1}{\cos {\left (x \right )}} \right )}}{\cos {\left (x \right )}}\, dx
\]