49.19.1 problem 1(i)
Internal
problem
ID
[7721]
Book
:
An
introduction
to
Ordinary
Differential
Equations.
Earl
A.
Coddington.
Dover.
NY
1961
Section
:
Chapter
4.
Linear
equations
with
Regular
Singular
Points.
Page
166
Problem
number
:
1(i)
Date
solved
:
Wednesday, March 05, 2025 at 04:51:19 AM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} 2 x^{2} y^{\prime \prime }+\left (x^{2}+5 x \right ) y^{\prime }+\left (x^{2}-2\right ) y&=0 \end{align*}
Using series method with expansion around
\begin{align*} 0 \end{align*}
✓ Maple. Time used: 0.016 (sec). Leaf size: 56
Order:=8;
ode:=2*x^2*diff(diff(y(x),x),x)+(x^2+5*x)*diff(y(x),x)+(x^2-2)*y(x) = 0;
dsolve(ode,y(x),type='series',x=0);
\[
y = \frac {c_{2} x^{{5}/{2}} \left (1-\frac {1}{14} x -\frac {25}{504} x^{2}+\frac {197}{33264} x^{3}+\frac {1921}{3459456} x^{4}-\frac {11653}{103783680} x^{5}+\frac {12923}{21171870720} x^{6}+\frac {917285}{1126343522304} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{1} \left (1-\frac {2}{3} x +\frac {5}{6} x^{2}+\frac {2}{9} x^{3}-\frac {19}{216} x^{4}-\frac {1}{540} x^{5}+\frac {101}{45360} x^{6}-\frac {4}{35721} x^{7}+\operatorname {O}\left (x^{8}\right )\right )}{x^{2}}
\]
✓ Mathematica. Time used: 0.007 (sec). Leaf size: 116
ode=2*x^2*D[y[x],{x,2}]+(5*x+x^2)*D[y[x],x]+(x^2-2)*y[x]==0;
ic={};
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
\[
y(x)\to c_1 \sqrt {x} \left (\frac {917285 x^7}{1126343522304}+\frac {12923 x^6}{21171870720}-\frac {11653 x^5}{103783680}+\frac {1921 x^4}{3459456}+\frac {197 x^3}{33264}-\frac {25 x^2}{504}-\frac {x}{14}+1\right )+\frac {c_2 \left (-\frac {4 x^7}{35721}+\frac {101 x^6}{45360}-\frac {x^5}{540}-\frac {19 x^4}{216}+\frac {2 x^3}{9}+\frac {5 x^2}{6}-\frac {2 x}{3}+1\right )}{x^2}
\]
✓ Sympy. Time used: 1.160 (sec). Leaf size: 116
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(2*x**2*Derivative(y(x), (x, 2)) + (x**2 - 2)*y(x) + (x**2 + 5*x)*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
\[
y{\left (x \right )} = C_{2} \sqrt {x} \left (\frac {12923 x^{6}}{21171870720} - \frac {11653 x^{5}}{103783680} + \frac {1921 x^{4}}{3459456} + \frac {197 x^{3}}{33264} - \frac {25 x^{2}}{504} - \frac {x}{14} + 1\right ) + \frac {C_{1} \left (\frac {2579 x^{9}}{1337394240} - \frac {433 x^{8}}{22861440} - \frac {4 x^{7}}{35721} + \frac {101 x^{6}}{45360} - \frac {x^{5}}{540} - \frac {19 x^{4}}{216} + \frac {2 x^{3}}{9} + \frac {5 x^{2}}{6} - \frac {2 x}{3} + 1\right )}{x^{2}} + O\left (x^{8}\right )
\]