49.18.2 problem 1(b)

Internal problem ID [7719]
Book : An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY 1961
Section : Chapter 4. Linear equations with Regular Singular Points. Page 159
Problem number : 1(b)
Date solved : Sunday, March 30, 2025 at 12:19:51 PM
CAS classification : [_Lienard]

\begin{align*} x^{2} y^{\prime \prime }+x y^{\prime }+x^{2} y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Maple. Time used: 0.033 (sec). Leaf size: 36

Order:=8; 
ode:=x^2*diff(diff(y(x),x),x)+x*diff(y(x),x)+x^2*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (c_2 \ln \left (x \right )+c_1 \right ) \left (1-\frac {1}{4} x^{2}+\frac {1}{64} x^{4}-\frac {1}{2304} x^{6}+\operatorname {O}\left (x^{8}\right )\right )+\left (\frac {1}{4} x^{2}-\frac {3}{128} x^{4}+\frac {11}{13824} x^{6}+\operatorname {O}\left (x^{8}\right )\right ) c_2 \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 81

ode=x^2*D[y[x],{x,2}]+x*D[y[x],x]+x^2*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_1 \left (-\frac {x^6}{2304}+\frac {x^4}{64}-\frac {x^2}{4}+1\right )+c_2 \left (\frac {11 x^6}{13824}-\frac {3 x^4}{128}+\frac {x^2}{4}+\left (-\frac {x^6}{2304}+\frac {x^4}{64}-\frac {x^2}{4}+1\right ) \log (x)\right ) \]

✓ Sympy. Time used: 0.810 (sec). Leaf size: 24

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*y(x) + x**2*Derivative(y(x), (x, 2)) + x*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
 

\[ y{\left (x \right )} = C_{1} \left (- \frac {x^{6}}{2304} + \frac {x^{4}}{64} - \frac {x^{2}}{4} + 1\right ) + O\left (x^{8}\right ) \]