49.17.2 problem 1(b)
Internal
problem
ID
[7709]
Book
:
An
introduction
to
Ordinary
Differential
Equations.
Earl
A.
Coddington.
Dover.
NY
1961
Section
:
Chapter
4.
Linear
equations
with
Regular
Singular
Points.
Page
154
Problem
number
:
1(b)
Date
solved
:
Wednesday, March 05, 2025 at 04:51:03 AM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} 3 x^{2} y^{\prime \prime }+x^{6} y^{\prime }+2 x y&=0 \end{align*}
Using series method with expansion around
\begin{align*} 0 \end{align*}
✓ Maple. Time used: 0.017 (sec). Leaf size: 70
Order:=8;
ode:=3*x^2*diff(diff(y(x),x),x)+x^6*diff(y(x),x)+2*x*y(x) = 0;
dsolve(ode,y(x),type='series',x=0);
\[
y = c_{1} x \left (1-\frac {1}{3} x +\frac {1}{27} x^{2}-\frac {1}{486} x^{3}+\frac {1}{14580} x^{4}-\frac {7291}{656100} x^{5}+\frac {225991}{41334300} x^{6}-\frac {2522341}{3472081200} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (-\frac {2}{3} x +\frac {2}{9} x^{2}-\frac {2}{81} x^{3}+\frac {1}{729} x^{4}-\frac {1}{21870} x^{5}+\frac {7291}{984150} x^{6}-\frac {225991}{62001450} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (1-\frac {1}{3} x^{2}+\frac {14}{243} x^{3}-\frac {35}{8748} x^{4}+\frac {101}{656100} x^{5}+\frac {69199}{14762250} x^{6}+\frac {19882543}{4340101500} x^{7}+\operatorname {O}\left (x^{8}\right )\right )\right )
\]
✓ Mathematica. Time used: 0.041 (sec). Leaf size: 121
ode=3*x^2*D[y[x],{x,2}]+x^6*D[y[x],x]+2*x*y[x]==0;
ic={};
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
\[
y(x)\to c_1 \left (\frac {x \left (7291 x^5-45 x^4+1350 x^3-24300 x^2+218700 x-656100\right ) \log (x)}{984150}+\frac {-80332 x^6+5895 x^5-158625 x^4+2430000 x^3-16402500 x^2+19683000 x+29524500}{29524500}\right )+c_2 \left (\frac {225991 x^7}{41334300}-\frac {7291 x^6}{656100}+\frac {x^5}{14580}-\frac {x^4}{486}+\frac {x^3}{27}-\frac {x^2}{3}+x\right )
\]
✓ Sympy. Time used: 1.091 (sec). Leaf size: 42
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x**6*Derivative(y(x), x) + 3*x**2*Derivative(y(x), (x, 2)) + 2*x*y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
\[
y{\left (x \right )} = C_{1} x \left (\frac {225991 x^{6}}{41334300} - \frac {7291 x^{5}}{656100} + \frac {x^{4}}{14580} - \frac {x^{3}}{486} + \frac {x^{2}}{27} - \frac {x}{3} + 1\right ) + O\left (x^{8}\right )
\]