84.34.5 problem 20.18

Internal problem ID [22332]
Book : Schaums outline series. Differential Equations By Richard Bronson. 1973. McGraw-Hill Inc. ISBN 0-07-008009-7
Section : Chapter 20. Regular singular points and the method of Frobenius. Supplementary problems
Problem number : 20.18
Date solved : Thursday, October 02, 2025 at 08:37:38 PM
CAS classification : [[_Emden, _Fowler]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 28

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

\[ y = \left (c_2 \ln \left (x \right )+c_1 \right ) \left (1-\frac {1}{9} x^{3}+\operatorname {O}\left (x^{6}\right )\right )+\left (\frac {2}{27} x^{3}+\operatorname {O}\left (x^{6}\right )\right ) c_2 \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 39

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

\[ y(x)\to c_1 \left (1-\frac {x^3}{9}\right )+c_2 \left (\frac {2 x^3}{27}+\left (1-\frac {x^3}{9}\right ) \log (x)\right ) \]

✓ Sympy. Time used: 0.261 (sec). Leaf size: 14

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

\[ y{\left (x \right )} = C_{1} \left (1 - \frac {x^{3}}{9}\right ) + O\left (x^{6}\right ) \]