82.1.14 problem 23-16

Internal problem ID [21756]
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 : 23-16
Date solved : Thursday, October 02, 2025 at 08:01:51 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} x^{2} y^{\prime \prime }+4 x y^{\prime }+2 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.011 (sec). Leaf size: 18

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

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

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 16

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}]
 

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

✓ Sympy. Time used: 0.207 (sec). Leaf size: 14

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) + 4*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)
 

\[ y{\left (x \right )} = \frac {C_{2}}{x} + \frac {C_{1}}{x^{2}} + O\left (x^{6}\right ) \]