82.3.27 problem 25-27

Internal problem ID [21810]
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 : 25-27
Date solved : Thursday, October 02, 2025 at 08:02:29 PM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} 3 x^{2} y^{\prime \prime }-x y^{\prime }+y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.009 (sec). Leaf size: 17

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

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

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 16

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

\[ y(x)\to c_1 x+c_2 \sqrt [3]{x} \]

✓ Sympy. Time used: 0.202 (sec). Leaf size: 15

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

\[ y{\left (x \right )} = C_{2} x + C_{1} \sqrt [3]{x} + O\left (x^{6}\right ) \]