73.23.34 problem 33.11 (h)
Internal
problem
ID
[15609]
Book
:
Ordinary
Differential
Equations.
An
introduction
to
the
fundamentals.
Kenneth
B.
Howell.
second
edition.
CRC
Press.
FL,
USA.
2020
Section
:
Chapter
33.
Power
series
solutions
I:
Basic
computational
methods.
Additional
Exercises.
page
641
Problem
number
:
33.11
(h)
Date
solved
:
Monday, March 31, 2025 at 01:42:18 PM
CAS
classification
:
[_quadrature]
\begin{align*} y^{\prime }+\cos \left (y\right )&=0 \end{align*}
Using series method with expansion around
\begin{align*} 0 \end{align*}
✓ Maple. Time used: 0.008 (sec). Leaf size: 59
Order:=5;
ode:=diff(y(x),x)+cos(y(x)) = 0;
dsolve(ode,y(x),type='series',x=0);
\[
y = y \left (0\right )-\cos \left (y \left (0\right )\right ) x -\frac {\sin \left (2 y \left (0\right )\right ) x^{2}}{4}+\frac {\cos \left (y \left (0\right )\right ) \cos \left (2 y \left (0\right )\right ) x^{3}}{6}+\left (\frac {\sin \left (4 y \left (0\right )\right )}{32}+\frac {\sin \left (2 y \left (0\right )\right )}{24}\right ) x^{4}+O\left (x^{5}\right )
\]
✓ Mathematica. Time used: 0.116 (sec). Leaf size: 194
ode=D[y[x],x]+Cos[y[x]]==0;
ic={};
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,4}]
\[
y(x)\to 2 \arctan \left (\tanh \left (\frac {c_1}{2}\right )\right )+\frac {x^4 \left (-5 \tanh ^7\left (\frac {c_1}{2}\right )+19 \tanh ^5\left (\frac {c_1}{2}\right )-19 \tanh ^3\left (\frac {c_1}{2}\right )+5 \tanh \left (\frac {c_1}{2}\right )\right )}{12 \left (1+\tanh ^2\left (\frac {c_1}{2}\right )\right ){}^4}+\frac {x^3 \left (1-\tanh ^6\left (\frac {c_1}{2}\right )+7 \tanh ^4\left (\frac {c_1}{2}\right )-7 \tanh ^2\left (\frac {c_1}{2}\right )\right )}{6 \left (1+\tanh ^2\left (\frac {c_1}{2}\right )\right ){}^3}+\frac {x^2 \left (\tanh ^3\left (\frac {c_1}{2}\right )-\tanh \left (\frac {c_1}{2}\right )\right )}{\left (1+\tanh ^2\left (\frac {c_1}{2}\right )\right ){}^2}+\frac {x \left (-1+\tanh ^2\left (\frac {c_1}{2}\right )\right )}{1+\tanh ^2\left (\frac {c_1}{2}\right )}
\]
✓ Sympy. Time used: 1.064 (sec). Leaf size: 66
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(cos(y(x)) + Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics,hint="1st_power_series",x0=0,n=5)
\[
y{\left (x \right )} = - x \cos {\left (C_{1} \right )} - \frac {x^{2} \sin {\left (C_{1} \right )} \cos {\left (C_{1} \right )}}{2} - \frac {x^{3} \left (\sin ^{2}{\left (C_{1} \right )} - \cos ^{2}{\left (C_{1} \right )}\right ) \cos {\left (C_{1} \right )}}{6} - \frac {x^{4} \left (\sin ^{2}{\left (C_{1} \right )} - 5 \cos ^{2}{\left (C_{1} \right )}\right ) \sin {\left (C_{1} \right )} \cos {\left (C_{1} \right )}}{24} + C_{1} + O\left (x^{5}\right )
\]