52.2.2 problem 2
Internal
problem
ID
[8253]
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
:
2
Date
solved
:
Wednesday, March 05, 2025 at 05:33:18 AM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} x \left (x +3\right )^{2} y^{\prime \prime }-y&=0 \end{align*}
Using series method with expansion around
\begin{align*} 0 \end{align*}
✓ Maple. Time used: 0.020 (sec). Leaf size: 70
Order:=8;
ode:=x*(x+3)^2*diff(diff(y(x),x),x)-y(x) = 0;
dsolve(ode,y(x),type='series',x=0);
\[
y = c_{1} x \left (1+\frac {1}{18} x -\frac {11}{972} x^{2}+\frac {277}{104976} x^{3}-\frac {12539}{18895680} x^{4}+\frac {893821}{5101833600} x^{5}-\frac {13183337}{275499014400} x^{6}+\frac {265861081}{19835929036800} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (\frac {1}{9} x +\frac {1}{162} x^{2}-\frac {11}{8748} x^{3}+\frac {277}{944784} x^{4}-\frac {12539}{170061120} x^{5}+\frac {893821}{45916502400} x^{6}-\frac {13183337}{2479491129600} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (1-\frac {5}{108} x^{2}+\frac {167}{26244} x^{3}-\frac {13583}{11337408} x^{4}+\frac {1327279}{5101833600} x^{5}-\frac {21146863}{344373768000} x^{6}+\frac {379766273}{24794911296000} x^{7}+\operatorname {O}\left (x^{8}\right )\right )\right )
\]
✓ Mathematica. Time used: 0.749 (sec). Leaf size: 121
ode=x*(x+3)^2*D[y[x],{x,2}]-y[x]==0;
ic={};
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
\[
y(x)\to c_1 \left (\frac {x \left (893821 x^5-3385530 x^4+13462200 x^3-57736800 x^2+283435200 x+5101833600\right ) \log (x)}{45916502400}+\frac {24742849 x^6-74732085 x^5+184497750 x^4+52488000 x^3-10628820000 x^2+382637520000 x+688747536000}{688747536000}\right )+c_2 \left (-\frac {13183337 x^7}{275499014400}+\frac {893821 x^6}{5101833600}-\frac {12539 x^5}{18895680}+\frac {277 x^4}{104976}-\frac {11 x^3}{972}+\frac {x^2}{18}+x\right )
\]
✓ Sympy. Time used: 1.011 (sec). Leaf size: 39
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x*(x + 3)**2*Derivative(y(x), (x, 2)) - y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
\[
y{\left (x \right )} = C_{1} x \left (\frac {x^{6}}{3628800} + \frac {x^{5}}{86400} + \frac {x^{4}}{2880} + \frac {x^{3}}{144} + \frac {x^{2}}{12} + \frac {x}{2} + 1\right ) + O\left (x^{8}\right )
\]