82.3.17 problem 25-17

Internal problem ID [21800]
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 : 25-17
Date solved : Thursday, October 02, 2025 at 08:02:22 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✗ Maple

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

\[ \text {No solution found} \]

✓ Mathematica. Time used: 0.018 (sec). Leaf size: 90

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

\[ y(x)\to c_2 e^{-4/x} \left (-\frac {40885 x^5}{6144}+\frac {5525 x^4}{1536}-\frac {425 x^3}{192}+\frac {25 x^2}{16}-\frac {5 x}{4}+1\right ) x^3+c_1 \left (\frac {5 x^5}{3072}+\frac {5 x^4}{1536}+\frac {x^3}{96}+\frac {x^2}{16}+\frac {x}{2}+1\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) - (x + 4)*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**2*Derivative(y(x), (x, 2)) - (x + 4)*Derivative(y(x), x) + 2*y(x) does not match hint 2nd_power_series_regular