50.4.53 problem 50

Internal problem ID [10228]
Book : Own collection of miscellaneous problems
Section : section 4.0
Problem number : 50
Date solved : Tuesday, September 30, 2025 at 07:12:53 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }-x y^{\prime }+\left (x^{2}-8\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.025 (sec). Leaf size: 35

Order:=6; 
ode:=x^2*diff(diff(y(x),x),x)-x*diff(y(x),x)+(x^2-8)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \,x^{4} \left (1-\frac {1}{16} x^{2}+\frac {1}{640} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_2 \left (-86400-10800 x^{2}-1350 x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{2}} \]

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 42

ode=x^2*D[y[x],{x,2}]-x*D[y[x],x]+(x^2-8)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (\frac {x^2}{64}+\frac {1}{x^2}+\frac {1}{8}\right )+c_2 \left (\frac {x^8}{640}-\frac {x^6}{16}+x^4\right ) \]

✓ Sympy. Time used: 0.251 (sec). Leaf size: 10

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) - x*Derivative(y(x), x) + (x**2 - 8)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{1} x^{4} + O\left (x^{6}\right ) \]