78.19.18 problem 7 (a)
Internal
problem
ID
[18437]
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
29.
Regular
singular
Points.
Problems
at
page
227
Problem
number
:
7
(a)
Date
solved
:
Tuesday, January 28, 2025 at 11:50:02 AM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} y^{\prime \prime }+\frac {n y^{\prime }}{x^{2}}+\frac {q y}{x^{3}}&=0 \end{align*}
Using series method with expansion around
\begin{align*} 0 \end{align*}
✗ Solution by Maple
Order:=6;
dsolve(diff(y(x),x$2)+n/x^2*diff(y(x),x)+q/x^3*y(x)=0,y(x),type='series',x=0);
\[ \text {No solution found} \]
✓ Solution by Mathematica
Time used: 0.756 (sec). Leaf size: 790
AsymptoticDSolveValue[D[y[x],{x,2}]+n/x^2*D[y[x],x]+q/x^3*y[x]==0,y[x],{x,0,"6"-1}]
\[
y(x)\to e^{n/x} \left (\frac {n}{x}\right )^{-\frac {q}{n}} \left (-\frac {c_2 x^5 (n+q) (2 n+q)^2 (3 n+q)^2 (4 n+q) \left (-\sin \left (\frac {\pi (n+q)}{n}\right ) \log \left (-\frac {n}{x}\right )+\sin \left (\frac {\pi (n+q)}{n}\right ) \log \left (\frac {n}{x}\right )+\pi \cos \left (\frac {\pi (n+q)}{n}\right )\right )}{6 \pi n^{11}}-\frac {c_2 x^4 (n+q) (2 n+q)^2 (3 n+q) \left (-\sin \left (\frac {\pi (n+q)}{n}\right ) \log \left (-\frac {n}{x}\right )+\sin \left (\frac {\pi (n+q)}{n}\right ) \log \left (\frac {n}{x}\right )+\pi \cos \left (\frac {\pi (n+q)}{n}\right )\right )}{2 \pi n^8}-\frac {c_2 x^3 (n+q) (2 n+q) \left (-\sin \left (\frac {\pi (n+q)}{n}\right ) \log \left (-\frac {n}{x}\right )+\sin \left (\frac {\pi (n+q)}{n}\right ) \log \left (\frac {n}{x}\right )+\pi \cos \left (\frac {\pi (n+q)}{n}\right )\right )}{\pi n^5}-\frac {c_2 x^2 \left (-\sin \left (\frac {\pi (n+q)}{n}\right ) \log \left (-\frac {n}{x}\right )+\sin \left (\frac {\pi (n+q)}{n}\right ) \log \left (\frac {n}{x}\right )+\pi \cos \left (\frac {\pi (n+q)}{n}\right )\right )}{\pi n^2}\right )+\left (-\frac {n}{x}\right )^{\frac {q}{n}} \left (\frac {c_1 q^2 x^5 (n-q)^2 (2 n-q)^2 (3 n-q)^2 (4 n-q) (n+q)}{120 n^{15} \operatorname {Gamma}\left (\frac {q}{n}+2\right )}-\frac {c_1 q^2 x^4 (n-q)^2 (2 n-q)^2 (3 n-q) (n+q)}{24 n^{12} \operatorname {Gamma}\left (\frac {q}{n}+2\right )}+\frac {c_1 q^2 x^3 (n-q)^2 (2 n-q) (n+q)}{6 n^9 \operatorname {Gamma}\left (\frac {q}{n}+2\right )}-\frac {c_1 q^2 x^2 (n-q) (n+q)}{2 n^6 \operatorname {Gamma}\left (\frac {q}{n}+2\right )}-\frac {c_1 q x (n+q)}{n^3 \operatorname {Gamma}\left (\frac {q}{n}+2\right )}+\frac {c_1}{\operatorname {Gamma}\left (\frac {q}{n}+2\right )}\right )+e^{n/x} \left (\frac {n}{x}\right )^{-\frac {q}{n}-2} \left (\frac {c_1 x^5 (n+q) (2 n+q)^2 (3 n+q)^2 (4 n+q)^2 (5 n+q)^2 (6 n+q)}{120 n^{15} \operatorname {Gamma}\left (-\frac {q}{n}\right )}+\frac {c_1 x^4 (n+q) (2 n+q)^2 (3 n+q)^2 (4 n+q)^2 (5 n+q)}{24 n^{12} \operatorname {Gamma}\left (-\frac {q}{n}\right )}+\frac {c_1 x^3 (n+q) (2 n+q)^2 (3 n+q)^2 (4 n+q)}{6 n^9 \operatorname {Gamma}\left (-\frac {q}{n}\right )}+\frac {c_1 x^2 (n+q) (2 n+q)^2 (3 n+q)}{2 n^6 \operatorname {Gamma}\left (-\frac {q}{n}\right )}+\frac {c_1 x (n+q) (2 n+q)}{n^3 \operatorname {Gamma}\left (-\frac {q}{n}\right )}+\frac {c_1}{\operatorname {Gamma}\left (-\frac {q}{n}\right )}\right )
\]