44.4.15 problem 4 (c)
Internal
problem
ID
[7028]
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
(c)
Date
solved
:
Wednesday, March 05, 2025 at 04:02:56 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 (3\right )&=3 \end{align*}
✓ Maple. Time used: 1.348 (sec). Leaf size: 107
ode:=diff(y(x),x) = sin(x)*cos(y(x));
ic:=y(3) = 3;
dsolve([ode,ic],y(x), singsol=all);
\[
y = \arctan \left (\frac {\sin \left (3\right ) {\mathrm e}^{2 \cos \left (3\right )-2 \cos \left (x \right )}+{\mathrm e}^{2 \cos \left (3\right )-2 \cos \left (x \right )}+\sin \left (3\right )-1}{\sin \left (3\right ) {\mathrm e}^{2 \cos \left (3\right )-2 \cos \left (x \right )}+{\mathrm e}^{2 \cos \left (3\right )-2 \cos \left (x \right )}-\sin \left (3\right )+1}, \frac {2 \cos \left (3\right ) {\mathrm e}^{\cos \left (3\right )-\cos \left (x \right )}}{\sin \left (3\right ) {\mathrm e}^{2 \cos \left (3\right )-2 \cos \left (x \right )}+{\mathrm e}^{2 \cos \left (3\right )-2 \cos \left (x \right )}-\sin \left (3\right )+1}\right )
\]
✓ Mathematica. Time used: 0.116 (sec). Leaf size: 27
ode=D[y[x],x]==Sin[x]*Cos[y[x]];
ic={y[3]==3};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
y(x)\to 2 \arctan \left (\tanh \left (\text {arctanh}\left (\tan \left (\frac {3}{2}\right )\right )-\frac {\cos (x)}{2}+\frac {\cos (3)}{2}\right )\right )
\]
✓ Sympy. Time used: 1.702 (sec). Leaf size: 138
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-sin(x)*cos(y(x)) + Derivative(y(x), x),0)
ics = {y(3): 3}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = \pi - \operatorname {asin}{\left (\frac {e^{2 \cos {\left (x \right )}} + \frac {\sin {\left (3 \right )} + 1}{- e^{- 2 \cos {\left (3 \right )}} + e^{- 2 \cos {\left (3 \right )}} \sin {\left (3 \right )}}}{- e^{2 \cos {\left (x \right )}} + \frac {\sin {\left (3 \right )} + 1}{- e^{- 2 \cos {\left (3 \right )}} + e^{- 2 \cos {\left (3 \right )}} \sin {\left (3 \right )}}} \right )}, \ y{\left (x \right )} = \operatorname {asin}{\left (\frac {e^{2 \cos {\left (x \right )}} + \frac {\sin {\left (3 \right )} + 1}{- e^{- 2 \cos {\left (3 \right )}} + e^{- 2 \cos {\left (3 \right )}} \sin {\left (3 \right )}}}{- e^{2 \cos {\left (x \right )}} + \frac {\sin {\left (3 \right )} + 1}{- e^{- 2 \cos {\left (3 \right )}} + e^{- 2 \cos {\left (3 \right )}} \sin {\left (3 \right )}}} \right )}\right ]
\]