73.24.27 problem 34.9 b(iii)
Internal
problem
ID
[15636]
Book
:
Ordinary
Differential
Equations.
An
introduction
to
the
fundamentals.
Kenneth
B.
Howell.
second
edition.
CRC
Press.
FL,
USA.
2020
Section
:
Chapter
34.
Power
series
solutions
II:
Generalization
and
theory.
Additional
Exercises.
page
678
Problem
number
:
34.9
b(iii)
Date
solved
:
Monday, March 31, 2025 at 01:43:36 PM
CAS
classification
:
[[_2nd_order, _exact, _linear, _homogeneous]]
\begin{align*} y^{\prime \prime }+\sin \left (x \right ) y^{\prime }+\cos \left (x \right ) y&=0 \end{align*}
Using series method with expansion around
\begin{align*} 0 \end{align*}
✓ Maple. Time used: 0.008 (sec). Leaf size: 44
Order:=7;
ode:=diff(diff(y(x),x),x)+sin(x)*diff(y(x),x)+cos(x)*y(x) = 0;
dsolve(ode,y(x),type='series',x=0);
\[
y = \left (1-\frac {1}{2} x^{2}+\frac {1}{6} x^{4}-\frac {31}{720} x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{3} x^{3}+\frac {1}{10} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{7}\right )
\]
✓ Mathematica. Time used: 0.001 (sec). Leaf size: 49
ode=D[y[x],{x,2}]+Sin[x]*D[y[x],x]+Cos[x]*y[x]==0;
ic={};
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,6}]
\[
y(x)\to c_2 \left (\frac {x^5}{10}-\frac {x^3}{3}+x\right )+c_1 \left (-\frac {31 x^6}{720}+\frac {x^4}{6}-\frac {x^2}{2}+1\right )
\]
✓ Sympy. Time used: 2.386 (sec). Leaf size: 167
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(y(x)*cos(x) + sin(x)*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=7)
\[
y{\left (x \right )} = C_{2} \left (\frac {x^{5} \sin ^{3}{\left (x \right )} \cos {\left (x \right )}}{120} - \frac {x^{5} \sin {\left (x \right )} \cos ^{2}{\left (x \right )}}{60} - \frac {x^{4} \sin ^{2}{\left (x \right )} \cos {\left (x \right )}}{24} + \frac {x^{4} \cos ^{2}{\left (x \right )}}{24} + \frac {x^{3} \sin {\left (x \right )} \cos {\left (x \right )}}{6} - \frac {x^{2} \cos {\left (x \right )}}{2} + 1\right ) + C_{1} x \left (\frac {x^{4} \sin ^{4}{\left (x \right )}}{120} - \frac {x^{4} \sin ^{2}{\left (x \right )} \cos {\left (x \right )}}{40} + \frac {x^{4} \cos ^{2}{\left (x \right )}}{120} - \frac {x^{3} \sin ^{3}{\left (x \right )}}{24} + \frac {x^{3} \sin {\left (x \right )} \cos {\left (x \right )}}{12} + \frac {x^{2} \sin ^{2}{\left (x \right )}}{6} - \frac {x^{2} \cos {\left (x \right )}}{6} - \frac {x \sin {\left (x \right )}}{2} + 1\right ) + O\left (x^{7}\right )
\]