64.15.3 problem 3

Internal problem ID [13509]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 6, Series solutions of linear differential equations. Section 6.2 (Frobenius). Exercises page 251
Problem number : 3
Date solved : Monday, March 31, 2025 at 07:59:58 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x^{4}-2 x^{3}+x^{2}\right ) y^{\prime \prime }+2 \left (x -1\right ) y^{\prime }+x^{2} y&=0 \end{align*}

Using series method with expansion around

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

✗ Maple

Order:=6; 
ode:=(x^4-2*x^3+x^2)*diff(diff(y(x),x),x)+2*(x-1)*diff(y(x),x)+x^2*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ \text {No solution found} \]

✓ Mathematica. Time used: 0.075 (sec). Leaf size: 71

ode=(x^4-2*x^3+x^2)*D[y[x],{x,2}]+2*(x-1)*D[y[x],x]+x^2*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (\frac {3 x^5}{10}+\frac {x^4}{4}+\frac {x^3}{6}+1\right )+c_2 e^{-2/x} \left (-\frac {429 x^5}{5}+\frac {91 x^4}{4}-\frac {31 x^3}{6}+3 x^2+1\right ) x^4 \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*y(x) + (2*x - 2)*Derivative(y(x), x) + (x**4 - 2*x**3 + x**2)*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

ValueError : ODE x**2*y(x) + (2*x - 2)*Derivative(y(x), x) + (x**4 - 2*x**3 + x**2)*Derivative(y(x), (x, 2)) does not match hint 2nd_power_series_regular