83.8.24 problem 25
Internal
problem
ID
[19079]
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.
Misc
examples
on
chapter
II
at
page
25
Problem
number
:
25
Date
solved
:
Monday, March 31, 2025 at 06:45:07 PM
CAS
classification
:
[[_homogeneous, `class C`], _dAlembert]
\begin{align*} y^{\prime }&=\sin \left (x +y\right )+\cos \left (x +y\right ) \end{align*}
✓ Maple. Time used: 0.017 (sec). Leaf size: 20
ode:=diff(y(x),x) = sin(x+y(x))+cos(x+y(x));
dsolve(ode,y(x), singsol=all);
\[
y = -x -2 \arctan \left (-\frac {{\mathrm e}^{x}}{c_1}+1\right )
\]
✓ Mathematica. Time used: 60.256 (sec). Leaf size: 387
ode=D[y[x],x]==Sin[x+y[x]]+Cos[x+y[x]];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to -2 \arccos \left (\frac {\sin \left (\frac {x}{2}\right )-\cos \left (\frac {x}{2}\right )-\sinh (c_1) \sin \left (\frac {x}{2}\right ) \sinh (x)-\cosh (c_1) \sin \left (\frac {x}{2}\right ) \sinh (x)-\sin \left (\frac {x}{2}\right ) \cosh (x) (\cosh (c_1)+\sinh (c_1))}{\sqrt {(3 \cosh (x+c_1)-\sinh (x+c_1)-2) (\cosh (x+c_1)+\sinh (x+c_1))}}\right ) \\
y(x)\to 2 \arccos \left (\frac {\sin \left (\frac {x}{2}\right )-\cos \left (\frac {x}{2}\right )-\sinh (c_1) \sin \left (\frac {x}{2}\right ) \sinh (x)-\cosh (c_1) \sin \left (\frac {x}{2}\right ) \sinh (x)-\sin \left (\frac {x}{2}\right ) \cosh (x) (\cosh (c_1)+\sinh (c_1))}{\sqrt {(3 \cosh (x+c_1)-\sinh (x+c_1)-2) (\cosh (x+c_1)+\sinh (x+c_1))}}\right ) \\
y(x)\to -2 \arccos \left (\frac {-\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )+\sinh (c_1) \sin \left (\frac {x}{2}\right ) \sinh (x)+\cosh (c_1) \sin \left (\frac {x}{2}\right ) \sinh (x)+\sin \left (\frac {x}{2}\right ) \cosh (x) (\cosh (c_1)+\sinh (c_1))}{\sqrt {(3 \cosh (x+c_1)-\sinh (x+c_1)-2) (\cosh (x+c_1)+\sinh (x+c_1))}}\right ) \\
y(x)\to 2 \arccos \left (\frac {-\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )+\sinh (c_1) \sin \left (\frac {x}{2}\right ) \sinh (x)+\cosh (c_1) \sin \left (\frac {x}{2}\right ) \sinh (x)+\sin \left (\frac {x}{2}\right ) \cosh (x) (\cosh (c_1)+\sinh (c_1))}{\sqrt {(3 \cosh (x+c_1)-\sinh (x+c_1)-2) (\cosh (x+c_1)+\sinh (x+c_1))}}\right ) \\
\end{align*}
✓ Sympy. Time used: 10.592 (sec). Leaf size: 102
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-sin(x + y(x)) - cos(x + y(x)) + Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
y{\left (x \right )} = - x + 2 \operatorname {acot}{\left (\frac {\sqrt {2 \sqrt {2} + 3} \left (e^{\sqrt {2 \sqrt {2} + 3} \left (C_{1} + x\right )} + e^{\sqrt {2} \sqrt {2 \sqrt {2} + 3} \left (C_{1} + x\right )}\right )}{e^{\sqrt {2 \sqrt {2} + 3} \left (C_{1} + x\right )} - e^{\sqrt {2} \sqrt {2 \sqrt {2} + 3} \left (C_{1} + x\right )}} \right )} - \frac {3 \pi }{4}
\]