83.5.20 problem 20
Internal
problem
ID
[19036]
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
(D)
at
page
16
Problem
number
:
20
Date
solved
:
Thursday, March 13, 2025 at 01:25:40 PM
CAS
classification
:
[_Bernoulli]
\begin{align*} 2 y^{\prime }-y \sec \left (x \right )&=y^{3} \tan \left (x \right ) \end{align*}
✓ Maple. Time used: 0.038 (sec). Leaf size: 78
ode:=2*diff(y(x),x)-y(x)*sec(x) = y(x)^3*tan(x);
dsolve(ode,y(x), singsol=all);
\begin{align*}
y \left (x \right ) &= \frac {\sqrt {-\cos \left (x \right ) \left (\cos \left (x \right )+\left (-1+\sin \left (x \right )\right ) \left (x +c_{1} \right )\right )}}{\left (x +c_{1} \right ) \sin \left (x \right )+\cos \left (x \right )-c_{1} -x} \\
y \left (x \right ) &= -\frac {\sqrt {-\cos \left (x \right ) \left (\cos \left (x \right )+\left (-1+\sin \left (x \right )\right ) \left (x +c_{1} \right )\right )}}{\left (x +c_{1} \right ) \sin \left (x \right )+\cos \left (x \right )-c_{1} -x} \\
\end{align*}
✓ Mathematica. Time used: 0.448 (sec). Leaf size: 136
ode=2*D[y[x],x]-y[x]*Sec[x]==y[x]^3*Tan[x];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to -\frac {i e^{\text {arctanh}\left (\tan \left (\frac {x}{2}\right )\right )} \sqrt {\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )}}{\sqrt {(x+2+c_1) \sin \left (\frac {x}{2}\right )-(x+c_1) \cos \left (\frac {x}{2}\right )}} \\
y(x)\to \frac {i e^{\text {arctanh}\left (\tan \left (\frac {x}{2}\right )\right )} \sqrt {\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )}}{\sqrt {(x+2+c_1) \sin \left (\frac {x}{2}\right )-(x+c_1) \cos \left (\frac {x}{2}\right )}} \\
y(x)\to 0 \\
\end{align*}
✓ Sympy. Time used: 20.559 (sec). Leaf size: 94
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-y(x)**3*tan(x) - y(x)/cos(x) + 2*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = - \sqrt {\frac {\sqrt {\sin {\left (x \right )} + 1}}{\left (C_{1} - \int \frac {\sqrt {\sin {\left (x \right )} + 1} \tan {\left (x \right )}}{\sqrt {\sin {\left (x \right )} - 1}}\, dx\right ) \sqrt {\sin {\left (x \right )} - 1}}}, \ y{\left (x \right )} = \sqrt {\frac {\sqrt {\sin {\left (x \right )} + 1}}{\left (C_{1} - \int \frac {\sqrt {\sin {\left (x \right )} + 1} \tan {\left (x \right )}}{\sqrt {\sin {\left (x \right )} - 1}}\, dx\right ) \sqrt {\sin {\left (x \right )} - 1}}}\right ]
\]