52.2.20 problem 20
Internal
problem
ID
[8271]
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
:
20
Date
solved
:
Wednesday, March 05, 2025 at 05:33:45 AM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} x^{2} y^{\prime \prime }-\left (x -\frac {2}{9}\right ) y&=0 \end{align*}
Using series method with expansion around
\begin{align*} 0 \end{align*}
✓ Maple. Time used: 0.012 (sec). Leaf size: 55
Order:=8;
ode:=x^2*diff(diff(y(x),x),x)-(x-2/9)*y(x) = 0;
dsolve(ode,y(x),type='series',x=0);
\[
y = c_{1} x^{{1}/{3}} \left (1+\frac {3}{2} x +\frac {9}{20} x^{2}+\frac {9}{160} x^{3}+\frac {27}{7040} x^{4}+\frac {81}{492800} x^{5}+\frac {81}{16755200} x^{6}+\frac {243}{2345728000} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} x^{{2}/{3}} \left (1+\frac {3}{4} x +\frac {9}{56} x^{2}+\frac {9}{560} x^{3}+\frac {27}{29120} x^{4}+\frac {81}{2329600} x^{5}+\frac {81}{88524800} x^{6}+\frac {243}{13632819200} x^{7}+\operatorname {O}\left (x^{8}\right )\right )
\]
✓ Mathematica. Time used: 0.005 (sec). Leaf size: 118
ode=x^2*D[y[x],{x,2}]-(x-2/9)*y[x]==0;
ic={};
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
\[
y(x)\to c_2 \sqrt [3]{x} \left (\frac {243 x^7}{2345728000}+\frac {81 x^6}{16755200}+\frac {81 x^5}{492800}+\frac {27 x^4}{7040}+\frac {9 x^3}{160}+\frac {9 x^2}{20}+\frac {3 x}{2}+1\right )+c_1 x^{2/3} \left (\frac {243 x^7}{13632819200}+\frac {81 x^6}{88524800}+\frac {81 x^5}{2329600}+\frac {27 x^4}{29120}+\frac {9 x^3}{560}+\frac {9 x^2}{56}+\frac {3 x}{4}+1\right )
\]
✓ Sympy. Time used: 1.038 (sec). Leaf size: 100
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x**2*Derivative(y(x), (x, 2)) - (x - 2/9)*y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
\[
y{\left (x \right )} = C_{2} x^{\frac {2}{3}} \left (\frac {81 x^{6}}{88524800} + \frac {81 x^{5}}{2329600} + \frac {27 x^{4}}{29120} + \frac {9 x^{3}}{560} + \frac {9 x^{2}}{56} + \frac {3 x}{4} + 1\right ) + C_{1} \sqrt [3]{x} \left (\frac {81 x^{6}}{16755200} + \frac {81 x^{5}}{492800} + \frac {27 x^{4}}{7040} + \frac {9 x^{3}}{160} + \frac {9 x^{2}}{20} + \frac {3 x}{2} + 1\right ) + O\left (x^{8}\right )
\]