78.22.2 problem 1 (b)
Internal
problem
ID
[18372]
Book
:
DIFFERENTIAL
EQUATIONS
WITH
APPLICATIONS
AND
HISTORICAL
NOTES
by
George
F.
Simmons.
3rd
edition.
2017.
CRC
press,
Boca
Raton
FL.
Section
:
Chapter
5.
Power
Series
Solutions
and
Special
Functions.
Section
32.
The
point
at
infinity.
Problems
at
page
244
Problem
number
:
1
(b)
Date
solved
:
Thursday, March 13, 2025 at 11:55:09 AM
CAS
classification
:
[_Bessel]
\begin{align*} x^{2} y^{\prime \prime }+x y^{\prime }+\left (-n^{2}+x^{2}\right ) y&=0 \end{align*}
Using series method with expansion around
\begin{align*} \infty \end{align*}
✗ Maple
Order:=6;
ode:=x^2*diff(diff(y(x),x),x)+x*diff(y(x),x)+(-n^2+x^2)*y(x) = 0;
dsolve(ode,y(x),type='series',x=infinity);
\[ \text {No solution found} \]
✓ Mathematica. Time used: 0.05 (sec). Leaf size: 534
ode=x^2*D[y[x],{x,2}]+x*D[y[x],x]+(x^2-n^2)*y[x]==0;
ic={};
AsymptoticDSolveValue[{ode,ic},y[x],{x,Infinity,5}]
\[
y(x)\to c_1 e^{-i x} \left (-\frac {i n^{10}}{3840 x^{11/2}}+\frac {n^8}{384 x^{9/2}}+\frac {11 i n^8}{1024 x^{11/2}}+\frac {i n^6}{48 x^{7/2}}-\frac {7 n^6}{128 x^{9/2}}-\frac {1463 i n^6}{10240 x^{11/2}}-\frac {n^4}{8 x^{5/2}}-\frac {35 i n^4}{192 x^{7/2}}+\frac {329 n^4}{1024 x^{9/2}}+\frac {17281 i n^4}{24576 x^{11/2}}-\frac {i n^2}{2 x^{3/2}}+\frac {5 n^2}{16 x^{5/2}}+\frac {259 i n^2}{768 x^{7/2}}-\frac {3229 n^2}{6144 x^{9/2}}-\frac {352407 i n^2}{327680 x^{11/2}}+\frac {i}{8 x^{3/2}}-\frac {9}{128 x^{5/2}}-\frac {75 i}{1024 x^{7/2}}+\frac {3675}{32768 x^{9/2}}+\frac {59535 i}{262144 x^{11/2}}+\frac {1}{\sqrt {x}}\right )+c_2 e^{i x} \left (\frac {i n^{10}}{3840 x^{11/2}}+\frac {n^8}{384 x^{9/2}}-\frac {11 i n^8}{1024 x^{11/2}}-\frac {i n^6}{48 x^{7/2}}-\frac {7 n^6}{128 x^{9/2}}+\frac {1463 i n^6}{10240 x^{11/2}}-\frac {n^4}{8 x^{5/2}}+\frac {35 i n^4}{192 x^{7/2}}+\frac {329 n^4}{1024 x^{9/2}}-\frac {17281 i n^4}{24576 x^{11/2}}+\frac {i n^2}{2 x^{3/2}}+\frac {5 n^2}{16 x^{5/2}}-\frac {259 i n^2}{768 x^{7/2}}-\frac {3229 n^2}{6144 x^{9/2}}+\frac {352407 i n^2}{327680 x^{11/2}}-\frac {i}{8 x^{3/2}}-\frac {9}{128 x^{5/2}}+\frac {75 i}{1024 x^{7/2}}+\frac {3675}{32768 x^{9/2}}-\frac {59535 i}{262144 x^{11/2}}+\frac {1}{\sqrt {x}}\right )
\]
✓ Sympy. Time used: 2.591 (sec). Leaf size: 367
from sympy import *
x = symbols("x")
n = symbols("n")
y = Function("y")
ode = Eq(x**2*Derivative(y(x), (x, 2)) + x*Derivative(y(x), x) + (-n**2 + x**2)*y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=inf,n=6)
\[
y{\left (x \right )} = - \frac {13 \left (C_{1} - x\right )^{5} r{\left (3 \right )}}{10 C_{1}^{2}} - \frac {n^{2} \left (C_{1} - x\right )^{5} r{\left (3 \right )}}{20 C_{1}^{2}} - \frac {5 \left (C_{1} - x\right )^{4} r{\left (3 \right )}}{4 C_{1}} + \frac {\left (C_{1} - x\right )^{5} r{\left (3 \right )}}{20} + C_{3} \left (\frac {7 C_{1} n^{4} \left (C_{1} - x\right )^{5}}{120} - \frac {7 C_{1} n^{2} \left (C_{1} - x\right )^{5}}{30} + \frac {\left (C_{1} - x\right )^{4}}{24} - \frac {\left (C_{1} - x\right )^{2}}{2} + 1 + \frac {\left (C_{1} - x\right )^{5}}{120 C_{1}} + \frac {\left (C_{1} - x\right )^{4}}{24 C_{1}} - \frac {\left (C_{1} - x\right )^{2}}{2 C_{1}} - \frac {n^{2} \left (C_{1} - x\right )^{4}}{12 C_{1}^{2}} + \frac {n^{2} \left (C_{1} - x\right )^{2}}{2 C_{1}^{2}} + \frac {\left (C_{1} - x\right )^{5}}{120 C_{1}^{2}} + \frac {\left (C_{1} - x\right )^{4}}{12 C_{1}^{2}} - \frac {n^{2} \left (C_{1} - x\right )^{5}}{15 C_{1}^{3}} - \frac {n^{2} \left (C_{1} - x\right )^{4}}{24 C_{1}^{3}} + \frac {7 \left (C_{1} - x\right )^{5}}{60 C_{1}^{3}} + \frac {\left (C_{1} - x\right )^{4}}{6 C_{1}^{3}} + \frac {n^{4} \left (C_{1} - x\right )^{4}}{24 C_{1}^{4}} - \frac {7 n^{2} \left (C_{1} - x\right )^{5}}{120 C_{1}^{4}} - \frac {n^{2} \left (C_{1} - x\right )^{4}}{6 C_{1}^{4}} + \frac {7 \left (C_{1} - x\right )^{5}}{30 C_{1}^{4}}\right ) + C_{2} \left (- C_{1} + x - \frac {\left (C_{1} - x\right )^{4}}{8 C_{1}} - \frac {\left (C_{1} - x\right )^{2}}{2 C_{1}} - \frac {7 \left (C_{1} - x\right )^{5}}{40 C_{1}^{2}} - \frac {n^{2} \left (C_{1} - x\right )^{4}}{24 C_{1}^{3}} + \frac {\left (C_{1} - x\right )^{4}}{6 C_{1}^{3}} - \frac {7 n^{2} \left (C_{1} - x\right )^{5}}{120 C_{1}^{4}} + \frac {7 \left (C_{1} - x\right )^{5}}{30 C_{1}^{4}}\right ) + O\left (x^{6}\right )
\]