20.25.11 problem 12

Internal problem ID [4016]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 11, Series Solutions to Linear Differential Equations. Exercises for 11.4. page 758
Problem number : 12
Date solved : Tuesday, March 04, 2025 at 05:22:47 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+x y^{\prime }-\left (2+x \right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.014 (sec). Leaf size: 169

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);
 

\[ y \left (x \right ) = c_{1} x^{-\sqrt {2}} \left (1+\frac {1}{1-2 \sqrt {2}} x +\frac {1}{20-12 \sqrt {2}} x^{2}-\frac {1}{228 \sqrt {2}-324} x^{3}+\frac {1}{8832-6240 \sqrt {2}} x^{4}-\frac {1}{480} \frac {1}{\left (2 \sqrt {2}-1\right ) \left (\sqrt {2}-1\right ) \left (-3+2 \sqrt {2}\right ) \left (-2+\sqrt {2}\right ) \left (-5+2 \sqrt {2}\right )} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x^{\sqrt {2}} \left (1+\frac {1}{1+2 \sqrt {2}} x +\frac {1}{20+12 \sqrt {2}} x^{2}+\frac {1}{228 \sqrt {2}+324} x^{3}+\frac {1}{8832+6240 \sqrt {2}} x^{4}+\frac {1}{244320 \sqrt {2}+345600} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 843

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

\begin{align*} \text {Solution too large to show}\end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) + x*Derivative(y(x), x) - (x + 2)*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 6 - x0