50.19.1 problem 1(a)

Internal problem ID [8109]
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 : 1(a)
Date solved : Sunday, March 30, 2025 at 12:45:36 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{3} \left (x -1\right ) y^{\prime \prime }-2 \left (x -1\right ) y^{\prime }+3 x y&=0 \end{align*}

Using series method with expansion around

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

✗ Maple

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

\[ \text {No solution found} \]

✓ Mathematica. Time used: 0.081 (sec). Leaf size: 108

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

\[ y(x)\to c_2 e^{-\frac {1}{x^2}} \left (\frac {1731 x^7}{320}-\frac {795 x^6}{128}-\frac {51 x^5}{40}+\frac {63 x^4}{32}+\frac {x^3}{2}-\frac {3 x^2}{4}+1\right ) x^3+c_1 \left (-\frac {51 x^7}{320}-\frac {19 x^6}{128}-\frac {9 x^5}{40}-\frac {9 x^4}{32}-\frac {x^3}{2}-\frac {3 x^2}{4}+1\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**3*(x - 1)*Derivative(y(x), (x, 2)) + 3*x*y(x) - (2*x - 2)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
 

ValueError : ODE x**3*(x - 1)*Derivative(y(x), (x, 2)) + 3*x*y(x) - (2*x - 2)*Derivative(y(x), x) does not match hint 2nd_power_series_regular