44.4.13 problem 4 (a)
Internal
problem
ID
[7026]
Book
:
A
First
Course
in
Differential
Equations
with
Modeling
Applications
by
Dennis
G.
Zill.
12
ed.
Metric
version.
2024.
Cengage
learning.
Section
:
Chapter
2.
First
order
differential
equations.
Section
2.1
Solution
curves
without
a
solution.
Exercises
2.1
at
page
44
Problem
number
:
4
(a)
Date
solved
:
Wednesday, March 05, 2025 at 04:02:46 AM
CAS
classification
:
[_separable]
\begin{align*} y^{\prime }&=\sin \left (x \right ) \cos \left (y\right ) \end{align*}
With initial conditions
\begin{align*} y \left (0\right )&=1 \end{align*}
✓ Maple. Time used: 0.703 (sec). Leaf size: 88
ode:=diff(y(x),x) = sin(x)*cos(y(x));
ic:=y(0) = 1;
dsolve([ode,ic],y(x), singsol=all);
\[
y = \arctan \left (\frac {\sin \left (1\right ) {\mathrm e}^{2-2 \cos \left (x \right )}+{\mathrm e}^{2-2 \cos \left (x \right )}+\sin \left (1\right )-1}{\sin \left (1\right ) {\mathrm e}^{2-2 \cos \left (x \right )}+{\mathrm e}^{2-2 \cos \left (x \right )}-\sin \left (1\right )+1}, \frac {2 \,{\mathrm e}^{1-\cos \left (x \right )} \cos \left (1\right )}{\sin \left (1\right ) {\mathrm e}^{2-2 \cos \left (x \right )}+{\mathrm e}^{2-2 \cos \left (x \right )}-\sin \left (1\right )+1}\right )
\]
✓ Mathematica. Time used: 0.727 (sec). Leaf size: 24
ode=D[y[x],x]==Sin[x]*Cos[y[x]];
ic={y[0]==1};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
y(x)\to 2 \arctan \left (\tanh \left (\text {arctanh}\left (\tan \left (\frac {1}{2}\right )\right )-\frac {\cos (x)}{2}+\frac {1}{2}\right )\right )
\]
✓ Sympy. Time used: 1.723 (sec). Leaf size: 97
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-sin(x)*cos(y(x)) + Derivative(y(x), x),0)
ics = {y(0): 1}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = \pi - \operatorname {asin}{\left (\frac {e^{2 \cos {\left (x \right )}} + \frac {e^{2} \sin {\left (1 \right )} + e^{2}}{-1 + \sin {\left (1 \right )}}}{- e^{2 \cos {\left (x \right )}} + \frac {e^{2} \sin {\left (1 \right )} + e^{2}}{-1 + \sin {\left (1 \right )}}} \right )}, \ y{\left (x \right )} = \operatorname {asin}{\left (\frac {e^{2 \cos {\left (x \right )}} + \frac {e^{2} \sin {\left (1 \right )} + e^{2}}{-1 + \sin {\left (1 \right )}}}{- e^{2 \cos {\left (x \right )}} + \frac {e^{2} \sin {\left (1 \right )} + e^{2}}{-1 + \sin {\left (1 \right )}}} \right )}\right ]
\]