89.7.5 problem 5
Internal
problem
ID
[24436]
Book
:
A
short
course
in
Differential
Equations.
Earl
D.
Rainville.
Second
edition.
1958.
Macmillan
Publisher,
NY.
CAT
58-5010
Section
:
Chapter
4.
Additional
topics
on
equations
of
first
order
and
first
degree.
Exercises
at
page
61
Problem
number
:
5
Date
solved
:
Thursday, October 02, 2025 at 10:31:53 PM
CAS
classification
:
[[_homogeneous, `class C`], _dAlembert]
\begin{align*} y^{\prime }&=\sin \left (x +y\right ) \end{align*}
✓ Maple. Time used: 0.010 (sec). Leaf size: 25
ode:=diff(y(x),x) = sin(x+y(x));
dsolve(ode,y(x), singsol=all);
\[
y = -x -2 \arctan \left (\frac {c_1 -x -2}{-x +c_1}\right )
\]
✓ Mathematica. Time used: 44.419 (sec). Leaf size: 541
ode=D[y[x],x]==Sin[x+y[x]];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to -2 \arccos \left (\frac {(x+c_1) \sin \left (\frac {x}{2}\right )-(x-2+c_1) \cos \left (\frac {x}{2}\right )}{\sqrt {2} \sqrt {x^2+2 (-1+c_1) x+2+c_1{}^2-2 c_1}}\right )\\ y(x)&\to 2 \arccos \left (\frac {(x+c_1) \sin \left (\frac {x}{2}\right )-(x-2+c_1) \cos \left (\frac {x}{2}\right )}{\sqrt {2} \sqrt {x^2+2 (-1+c_1) x+2+c_1{}^2-2 c_1}}\right )\\ y(x)&\to -2 \arccos \left (\frac {(x-2+c_1) \cos \left (\frac {x}{2}\right )-(x+c_1) \sin \left (\frac {x}{2}\right )}{\sqrt {2} \sqrt {x^2+2 (-1+c_1) x+2+c_1{}^2-2 c_1}}\right )\\ y(x)&\to 2 \arccos \left (\frac {(x-2+c_1) \cos \left (\frac {x}{2}\right )-(x+c_1) \sin \left (\frac {x}{2}\right )}{\sqrt {2} \sqrt {x^2+2 (-1+c_1) x+2+c_1{}^2-2 c_1}}\right )\\ y(x)&\to -2 \arccos \left (\frac {\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )}{\sqrt {2}}\right )\\ y(x)&\to 2 \arccos \left (\frac {\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )}{\sqrt {2}}\right )\\ y(x)&\to -2 \arccos \left (\frac {\sin \left (\frac {x}{2}\right )-\cos \left (\frac {x}{2}\right )}{\sqrt {2}}\right )\\ y(x)&\to 2 \arccos \left (\frac {\sin \left (\frac {x}{2}\right )-\cos \left (\frac {x}{2}\right )}{\sqrt {2}}\right )\\ y(x)&\to -2 \arccos \left (\frac {(x-2) \cos \left (\frac {x}{2}\right )-x \sin \left (\frac {x}{2}\right )}{\sqrt {2} \sqrt {x^2-2 x+2}}\right )\\ y(x)&\to 2 \arccos \left (\frac {(x-2) \cos \left (\frac {x}{2}\right )-x \sin \left (\frac {x}{2}\right )}{\sqrt {2} \sqrt {x^2-2 x+2}}\right )\\ y(x)&\to -2 \arccos \left (\frac {x \sin \left (\frac {x}{2}\right )-(x-2) \cos \left (\frac {x}{2}\right )}{\sqrt {2} \sqrt {x^2-2 x+2}}\right )\\ y(x)&\to 2 \arccos \left (\frac {x \sin \left (\frac {x}{2}\right )-(x-2) \cos \left (\frac {x}{2}\right )}{\sqrt {2} \sqrt {x^2-2 x+2}}\right ) \end{align*}
✓ Sympy. Time used: 1.048 (sec). Leaf size: 17
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-sin(x + y(x)) + Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
y{\left (x \right )} = - x - 2 \operatorname {atan}{\left (\frac {C_{1} + x + 2}{C_{1} + x} \right )}
\]