83.14.8 problem 2 (g)

Internal problem ID [22023]
Book : Differential Equations By Kaj L. Nielsen. Second edition 1966. Barnes and nobel. 66-28306
Section : Chapter X. Solution in power series. Ex. XVIII at page 174
Problem number : 2 (g)
Date solved : Thursday, October 02, 2025 at 08:21:46 PM
CAS classification : [[_Emden, _Fowler]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.009 (sec). Leaf size: 25

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

\[ y = c_1 x \left (1+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_2 \left (-144+\operatorname {O}\left (x^{6}\right )\right )}{x^{3}} \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 14

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

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

✓ Sympy. Time used: 0.211 (sec). Leaf size: 14

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

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