43.2.5 problem 7.3.7

Internal problem ID [6862]
Book : Notes on Diffy Qs. Differential Equations for Engineers. By by Jiri Lebl, 2013.
Section : Chapter 7. POWER SERIES METHODS. 7.3.2 The method of Frobenius. Exercises. page 300
Problem number : 7.3.7
Date solved : Sunday, March 30, 2025 at 11:24:43 AM
CAS classification : [[_Emden, _Fowler]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.033 (sec). Leaf size: 28

Order:=6; 
ode:=x^2*diff(diff(y(x),x),x)-x*diff(y(x),x)-y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = x \left (x^{-\sqrt {2}} c_1 +x^{\sqrt {2}} c_2 \right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 30

ode=x^2*D[y[x],{x,2}]-x*D[y[x],x]-y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 x^{1+\sqrt {2}}+c_2 x^{1-\sqrt {2}} \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) - x*Derivative(y(x), x) - y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

NotImplementedError : Not sure of sign of 5 - x0