82.3.32 problem 25-32

Internal problem ID [21815]
Book : The Differential Equations Problem Solver. VOL. II. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 25. Power series about a singular point. Page 762
Problem number : 25-32
Date solved : Thursday, October 02, 2025 at 08:02:32 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.009 (sec). Leaf size: 29

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

\[ y = \left (\left (\frac {1}{2}+\operatorname {O}\left (x^{3}\right )\right ) x^{2}+\left (1+\operatorname {O}\left (x^{6}\right )\right ) \left (c_1 x +c_2 \right )\right ) x \]

✓ Mathematica. Time used: 0.017 (sec). Leaf size: 21

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

\[ y(x)\to \frac {x^3}{2}+c_2 x^2+c_1 x \]

✗ Sympy

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

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