50.18.11 problem 5
Internal
problem
ID
[8105]
Book
:
Differential
Equations:
Theory,
Technique,
and
Practice
by
George
Simmons,
Steven
Krantz.
McGraw-Hill
NY.
2007.
1st
Edition.
Section
:
Chapter
4.
Power
Series
Solutions
and
Special
Functions.
Section
4.3.
Second-Order
Linear
Equations:
Ordinary
Points.
Page
169
Problem
number
:
5
Date
solved
:
Wednesday, March 05, 2025 at 05:29:56 AM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} y^{\prime \prime }+\left (p +\frac {1}{2}-\frac {x^{2}}{4}\right ) y&=0 \end{align*}
Using series method with expansion around
\begin{align*} 0 \end{align*}
✓ Maple. Time used: 0.003 (sec). Leaf size: 109
Order:=8;
ode:=diff(diff(y(x),x),x)+(p+1/2-1/4*x^2)*y(x) = 0;
dsolve(ode,y(x),type='series',x=0);
\[
y = \left (1-\frac {\left (2 p +1\right ) x^{2}}{4}+\frac {\left (4 p^{2}+4 p +3\right ) x^{4}}{96}-\frac {\left (8 p^{3}+12 p^{2}+34 p +15\right ) x^{6}}{5760}\right ) y \left (0\right )+\left (x -\frac {\left (2 p +1\right ) x^{3}}{12}+\frac {\left (4 p^{2}+4 p +7\right ) x^{5}}{480}-\frac {\left (8 p^{3}+12 p^{2}+58 p +27\right ) x^{7}}{40320}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right )
\]
✓ Mathematica. Time used: 0.002 (sec). Leaf size: 142
ode=D[y[x],{x,2}]+(p+1/2-x^2/4)*y[x]==0;
ic={};
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
\[
y(x)\to c_2 \left (\frac {(-4 p-2) (4 p+2)^2 x^7}{322560}+\frac {13 (-4 p-2) x^7}{40320}+\frac {(4 p+2)^2 x^5}{1920}+\frac {1}{24} (-4 p-2) x^3+\frac {x^5}{80}+x\right )+c_1 \left (\frac {(-4 p-2) (4 p+2)^2 x^6}{46080}+\frac {7 (-4 p-2) x^6}{5760}+\frac {1}{384} (4 p+2)^2 x^4+\frac {1}{8} (-4 p-2) x^2+\frac {x^4}{48}+1\right )
\]
✓ Sympy. Time used: 1.186 (sec). Leaf size: 143
from sympy import *
x = symbols("x")
p = symbols("p")
y = Function("y")
ode = Eq((p - x**2/4 + 1/2)*y(x) + Derivative(y(x), (x, 2)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=8)
\[
y{\left (x \right )} = - \frac {x^{5} r{\left (3 \right )}}{40} + \frac {x^{7} r{\left (3 \right )}}{160} - \frac {p x^{5} r{\left (3 \right )}}{20} + \frac {p x^{7} r{\left (3 \right )}}{840} + \frac {p^{2} x^{7} r{\left (3 \right )}}{840} + C_{2} \left (- \frac {p^{3} x^{6}}{720} - \frac {p^{2} x^{6}}{480} + \frac {p^{2} x^{4}}{24} - \frac {17 p x^{6}}{2880} + \frac {p x^{4}}{24} - \frac {p x^{2}}{2} - \frac {x^{6}}{384} + \frac {x^{4}}{32} - \frac {x^{2}}{4} + 1\right ) + C_{1} x \left (- \frac {p x^{6}}{3360} - \frac {x^{6}}{6720} + \frac {x^{4}}{80} + 1\right ) + O\left (x^{8}\right )
\]