77.3.1 problem 1
Internal
problem
ID
[20342]
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
:
1
Date
solved
:
Thursday, October 02, 2025 at 05:44:20 PM
CAS
classification
:
[_separable]
\begin{align*} x \cos \left (y\right )^{2}&=y \cos \left (x \right )^{2} y^{\prime } \end{align*}
✓ Maple. Time used: 0.062 (sec). Leaf size: 25
ode:=x*cos(y(x))^2 = y(x)*cos(x)^2*diff(y(x),x);
dsolve(ode,y(x), singsol=all);
\[
\tan \left (x \right ) x +\ln \left (\cos \left (x \right )\right )-y \tan \left (y\right )-\ln \left (\cos \left (y\right )\right )+c_1 = 0
\]
✓ Mathematica. Time used: 0.485 (sec). Leaf size: 55
ode=x*Cos[y[x]]^2==y[x]*Cos[x]^2*D[y[x],x];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \text {InverseFunction}\left [\frac {1}{2} (\text {$\#$1} \tan (\text {$\#$1})+\log (\cos (\text {$\#$1})))\&\right ]\left [\frac {1}{2} (x \tan (x)+\log (\cos (x)))+c_1\right ]\\ y(x)&\to -\frac {\pi }{2}\\ y(x)&\to \frac {\pi }{2} \end{align*}
✓ Sympy. Time used: 125.621 (sec). Leaf size: 316
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x*cos(y(x))**2 - y(x)*cos(x)**2*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\frac {2 x \tan {\left (\frac {x}{2} \right )}}{\tan ^{2}{\left (\frac {x}{2} \right )} - 1} - \frac {2 y{\left (x \right )} \tan {\left (\frac {y{\left (x \right )}}{2} \right )}}{\tan ^{2}{\left (\frac {y{\left (x \right )}}{2} \right )} - 1} + \frac {\log {\left (\tan {\left (\frac {y{\left (x \right )}}{2} \right )} - 1 \right )} \tan ^{2}{\left (\frac {y{\left (x \right )}}{2} \right )}}{\tan ^{2}{\left (\frac {y{\left (x \right )}}{2} \right )} - 1} - \frac {\log {\left (\tan {\left (\frac {y{\left (x \right )}}{2} \right )} - 1 \right )}}{\tan ^{2}{\left (\frac {y{\left (x \right )}}{2} \right )} - 1} + \frac {\log {\left (\tan {\left (\frac {y{\left (x \right )}}{2} \right )} + 1 \right )} \tan ^{2}{\left (\frac {y{\left (x \right )}}{2} \right )}}{\tan ^{2}{\left (\frac {y{\left (x \right )}}{2} \right )} - 1} - \frac {\log {\left (\tan {\left (\frac {y{\left (x \right )}}{2} \right )} + 1 \right )}}{\tan ^{2}{\left (\frac {y{\left (x \right )}}{2} \right )} - 1} - \frac {\log {\left (\tan ^{2}{\left (\frac {y{\left (x \right )}}{2} \right )} + 1 \right )} \tan ^{2}{\left (\frac {y{\left (x \right )}}{2} \right )}}{\tan ^{2}{\left (\frac {y{\left (x \right )}}{2} \right )} - 1} + \frac {\log {\left (\tan ^{2}{\left (\frac {y{\left (x \right )}}{2} \right )} + 1 \right )}}{\tan ^{2}{\left (\frac {y{\left (x \right )}}{2} \right )} - 1} - \frac {\log {\left (\tan {\left (\frac {x}{2} \right )} - 1 \right )} \tan ^{2}{\left (\frac {x}{2} \right )}}{\tan ^{2}{\left (\frac {x}{2} \right )} - 1} + \frac {\log {\left (\tan {\left (\frac {x}{2} \right )} - 1 \right )}}{\tan ^{2}{\left (\frac {x}{2} \right )} - 1} - \frac {\log {\left (\tan {\left (\frac {x}{2} \right )} + 1 \right )} \tan ^{2}{\left (\frac {x}{2} \right )}}{\tan ^{2}{\left (\frac {x}{2} \right )} - 1} + \frac {\log {\left (\tan {\left (\frac {x}{2} \right )} + 1 \right )}}{\tan ^{2}{\left (\frac {x}{2} \right )} - 1} + \frac {\log {\left (\tan ^{2}{\left (\frac {x}{2} \right )} + 1 \right )} \tan ^{2}{\left (\frac {x}{2} \right )}}{\tan ^{2}{\left (\frac {x}{2} \right )} - 1} - \frac {\log {\left (\tan ^{2}{\left (\frac {x}{2} \right )} + 1 \right )}}{\tan ^{2}{\left (\frac {x}{2} \right )} - 1} = C_{1}
\]