82.1.1 problem 23-6

Internal problem ID [21743]
Book : The Differential Equations Problem Solver. VOL. II. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 23. Power series. Page 695
Problem number : 23-6
Date solved : Thursday, October 02, 2025 at 08:01:42 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+2 y^{\prime }+y x&=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)+2*diff(y(x),x)+x*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ \text {No solution found} \]

✓ Mathematica. Time used: 0.02 (sec). Leaf size: 81

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

\[ y(x)\to c_2 e^{2/x} \left (\frac {13691 x^5}{480}+\frac {901 x^4}{96}+\frac {11 x^3}{3}+\frac {7 x^2}{4}+x+1\right ) x^2+c_1 \left (\frac {7 x^5}{240}-\frac {x^4}{32}+\frac {x^3}{12}-\frac {x^2}{4}+1\right ) \]

✗ Sympy

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