46.1.27 problem 25 expansion at 0
Internal
problem
ID
[9530]
Book
:
DIFFERENTIAL
EQUATIONS
with
Boundary
Value
Problems.
DENNIS
G.
ZILL,
WARREN
S.
WRIGHT,
MICHAEL
R.
CULLEN.
Brooks/Cole.
Boston,
MA.
2013.
8th
edition.
Section
:
CHAPTER
6
SERIES
SOLUTIONS
OF
LINEAR
EQUATIONS.
Section
6.2
SOLUTIONS
ABOUT
ORDINARY
POINTS.
EXERCISES
6.2.
Page
246
Problem
number
:
25
expansion
at
0
Date
solved
:
Tuesday, September 30, 2025 at 06:20:07 PM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} \cos \left (x \right ) y^{\prime \prime }+y^{\prime }+5 y&=0 \end{align*}
Using series method with expansion around
\begin{align*} 0 \end{align*}
✓ Maple. Time used: 0.009 (sec). Leaf size: 74
Order:=8;
ode:=cos(x)*diff(diff(y(x),x),x)+diff(y(x),x)+5*y(x) = 0;
dsolve(ode,y(x),type='series',x=0);
\[
y = \left (1-\frac {5}{2} x^{2}+\frac {5}{6} x^{3}+\frac {5}{8} x^{4}-\frac {5}{24} x^{5}+\frac {1}{16} x^{6}-\frac {13}{336} x^{7}\right ) y \left (0\right )+\left (x -\frac {1}{2} x^{2}-\frac {2}{3} x^{3}+\frac {1}{3} x^{4}+\frac {1}{80} x^{6}+\frac {11}{5040} x^{7}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right )
\]
✓ Mathematica. Time used: 0.002 (sec). Leaf size: 91
ode=Cos[x]*D[y[x],{x,2}]+D[y[x],x]+5*y[x]==0;
ic={};
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
\[
y(x)\to c_2 \left (\frac {11 x^7}{5040}+\frac {x^6}{80}+\frac {x^4}{3}-\frac {2 x^3}{3}-\frac {x^2}{2}+x\right )+c_1 \left (-\frac {13 x^7}{336}+\frac {x^6}{16}-\frac {5 x^5}{24}+\frac {5 x^4}{8}+\frac {5 x^3}{6}-\frac {5 x^2}{2}+1\right )
\]
✓ Sympy. Time used: 0.936 (sec). Leaf size: 230
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(5*y(x) + cos(x)*Derivative(y(x), (x, 2)) + Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=8)
\[
y{\left (x \right )} = C_{2} \left (- \frac {25 x^{6}}{144 \cos ^{3}{\left (x \right )}} + \frac {5 x^{6}}{48 \cos ^{4}{\left (x \right )}} - \frac {x^{6}}{144 \cos ^{5}{\left (x \right )}} - \frac {5 x^{5}}{12 \cos ^{3}{\left (x \right )}} + \frac {x^{5}}{24 \cos ^{4}{\left (x \right )}} + \frac {25 x^{4}}{24 \cos ^{2}{\left (x \right )}} - \frac {5 x^{4}}{24 \cos ^{3}{\left (x \right )}} + \frac {5 x^{3}}{6 \cos ^{2}{\left (x \right )}} - \frac {5 x^{2}}{2 \cos {\left (x \right )}} + 1\right ) + C_{1} x \left (- \frac {5 x^{5}}{48 \cos ^{3}{\left (x \right )}} + \frac {x^{5}}{36 \cos ^{4}{\left (x \right )}} - \frac {x^{5}}{720 \cos ^{5}{\left (x \right )}} + \frac {5 x^{4}}{24 \cos ^{2}{\left (x \right )}} - \frac {x^{4}}{8 \cos ^{3}{\left (x \right )}} + \frac {x^{4}}{120 \cos ^{4}{\left (x \right )}} + \frac {5 x^{3}}{12 \cos ^{2}{\left (x \right )}} - \frac {x^{3}}{24 \cos ^{3}{\left (x \right )}} - \frac {5 x^{2}}{6 \cos {\left (x \right )}} + \frac {x^{2}}{6 \cos ^{2}{\left (x \right )}} - \frac {x}{2 \cos {\left (x \right )}} + 1\right ) + O\left (x^{8}\right )
\]