37.2.10 problem 7.3.8 (e)

Internal problem ID [8153]
Book : Notes on Diffy Qs. Differential Equations for Engineers. By by Jiri Lebl, 2013.
Section : Chapter 7. POWER SERIES METHODS. 7.3.2 The method of Frobenius. Exercises. page 300
Problem number : 7.3.8 (e)
Date solved : Tuesday, September 30, 2025 at 05:16:12 PM
CAS classification : [[_2nd_order, _missing_x]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 49

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

\[ y = \left (1-\frac {1}{2} x^{2}+\frac {1}{6} x^{3}-\frac {1}{120} x^{5}\right ) y \left (0\right )+\left (x -\frac {1}{2} x^{2}+\frac {1}{24} x^{4}-\frac {1}{120} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 56

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

\[ y(x)\to c_2 \left (-\frac {x^5}{120}+\frac {x^4}{24}-\frac {x^2}{2}+x\right )+c_1 \left (-\frac {x^5}{120}+\frac {x^3}{6}-\frac {x^2}{2}+1\right ) \]

✓ Sympy. Time used: 0.223 (sec). Leaf size: 32

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

\[ y{\left (x \right )} = C_{2} \left (\frac {x^{3}}{6} - \frac {x^{2}}{2} + 1\right ) + C_{1} x \left (\frac {x^{3}}{24} - \frac {x}{2} + 1\right ) + O\left (x^{6}\right ) \]