65.13.8 problem 20.4

Internal problem ID [13735]
Book : AN INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS by JAMES C. ROBINSON. Cambridge University Press 2004
Section : Chapter 20, Series solutions of second order linear equations. Exercises page 195
Problem number : 20.4
Date solved : Wednesday, March 05, 2025 at 10:14:38 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\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: 0.047 (sec). Leaf size: 32

Order:=6; 
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);
 

\[ y = \left (\ln \left (x \right ) c_{2} +c_{1} \right ) \left (1+\frac {1}{4} x^{2}+\frac {1}{64} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (-\frac {1}{4} x^{2}-\frac {3}{128} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \]

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 60

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,0,5}]
 

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

✓ Sympy. Time used: 0.803 (sec). Leaf size: 19

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=6)
 

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