50.19.19 problem 6

Internal problem ID [8127]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 4. Power Series Solutions and Special Functions. Section 4.4. REGULAR SINGULAR POINTS. Page 175
Problem number : 6
Date solved : Sunday, March 30, 2025 at 12:46:08 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} y^{\prime \prime }+\frac {y^{\prime }}{x^{2}}-\frac {y}{x^{3}}&=0 \end{align*}

Using series method with expansion around

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

✗ Maple

Order:=8; 
ode:=diff(diff(y(x),x),x)+1/x^2*diff(y(x),x)-1/x^3*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ \text {No solution found} \]

✓ Mathematica. Time used: 0.035 (sec). Leaf size: 17

ode=D[y[x],{x,2}]+1/x^2*D[y[x],x]-1/x^3*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_1 x+c_2 e^{\frac {1}{x}} x \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), (x, 2)) + Derivative(y(x), x)/x**2 - y(x)/x**3,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
 

ValueError : ODE Derivative(y(x), (x, 2)) + Derivative(y(x), x)/x**2 - y(x)/x**3 does not match hint 2nd_power_series_regular