52.2.3 problem 3
Internal
problem
ID
[8254]
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.
6.3
SOLUTIONS
ABOUT
SINGULAR
POINTS.
EXERCISES
6.3.
Page
255
Problem
number
:
3
Date
solved
:
Sunday, March 30, 2025 at 12:49:57 PM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} \left (x^{2}-9\right )^{2} y^{\prime \prime }+\left (x +3\right ) y^{\prime }+2 y&=0 \end{align*}
Using series method with expansion around
\begin{align*} 0 \end{align*}
✓ Maple. Time used: 0.008 (sec). Leaf size: 79
Order:=8;
ode:=(x^2-9)^2*diff(diff(y(x),x),x)+(x+3)*diff(y(x),x)+2*y(x) = 0;
dsolve(ode,y(x),type='series',x=0);
\[
y = \left (1-\frac {1}{81} x^{2}+\frac {1}{6561} x^{3}-\frac {289}{708588} x^{4}+\frac {304}{23914845} x^{5}-\frac {194981}{7748409780} x^{6}+\frac {1732937}{1464449448420} x^{7}\right ) y \left (0\right )+\left (x -\frac {1}{54} x^{2}-\frac {13}{2187} x^{3}-\frac {131}{236196} x^{4}-\frac {596}{1594323} x^{5}-\frac {78469}{2582803260} x^{6}-\frac {13738871}{488149816140} x^{7}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right )
\]
✓ Mathematica. Time used: 0.004 (sec). Leaf size: 98
ode=(x^2-9)^2*D[y[x],{x,2}]+(x+3)*D[y[x],x]+2*y[x]==0;
ic={};
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
\[
y(x)\to c_1 \left (\frac {1732937 x^7}{1464449448420}-\frac {194981 x^6}{7748409780}+\frac {304 x^5}{23914845}-\frac {289 x^4}{708588}+\frac {x^3}{6561}-\frac {x^2}{81}+1\right )+c_2 \left (-\frac {13738871 x^7}{488149816140}-\frac {78469 x^6}{2582803260}-\frac {596 x^5}{1594323}-\frac {131 x^4}{236196}-\frac {13 x^3}{2187}-\frac {x^2}{54}+x\right )
\]
✓ Sympy. Time used: 1.135 (sec). Leaf size: 116
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq((x + 3)*Derivative(y(x), x) + (x**2 - 9)**2*Derivative(y(x), (x, 2)) + 2*y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=8)
\[
y{\left (x \right )} = - \frac {x^{4} r{\left (3 \right )}}{108} + \frac {232 x^{5} r{\left (3 \right )}}{3645} - \frac {1409 x^{6} r{\left (3 \right )}}{1180980} + \frac {1081877 x^{7} r{\left (3 \right )}}{223205220} + C_{2} \left (\frac {3617 x^{7}}{8135830269} - \frac {5377 x^{6}}{215233605} + \frac {8 x^{5}}{2657205} - \frac {8 x^{4}}{19683} - \frac {x^{2}}{81} + 1\right ) + C_{1} x \left (\frac {3617 x^{6}}{5423886846} - \frac {5377 x^{5}}{143489070} + \frac {4 x^{4}}{885735} - \frac {4 x^{3}}{6561} - \frac {x}{54} + 1\right ) + O\left (x^{8}\right )
\]