88.26.2 problem 2

Internal problem ID [24222]
Book : Elementary Differential Equations. By Lee Roy Wilcox and Herbert J. Curtis. 1961 first edition. International texbook company. Scranton, Penn. USA. CAT number 61-15976
Section : Chapter 7. Series Methods. Exercises at page 220
Problem number : 2
Date solved : Thursday, October 02, 2025 at 10:00:57 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }-5 x y^{\prime }+\left (-x^{2}+9\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.009 (sec). Leaf size: 36

Order:=6; 
ode:=x^2*diff(diff(y(x),x),x)-5*x*diff(y(x),x)+(-x^2+9)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (\left (c_2 \ln \left (x \right )+c_1 \right ) \left (1+\frac {1}{4} x^{2}+\frac {1}{64} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (-\frac {1}{4} x^{2}-\frac {3}{128} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_2 \right ) x^{3} \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 71

ode=x^2*D[y[x],{x,2}]-5*x*D[y[x],{x,1}]+(9-x^2)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (\frac {x^4}{64}+\frac {x^2}{4}+1\right ) x^3+c_2 \left (\left (-\frac {3 x^4}{128}-\frac {x^2}{4}\right ) x^3+\left (\frac {x^4}{64}+\frac {x^2}{4}+1\right ) x^3 \log (x)\right ) \]

✓ Sympy. Time used: 0.277 (sec). Leaf size: 17

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) - 5*x*Derivative(y(x), x) + (9 - x**2)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{1} x^{3} \left (\frac {x^{2}}{4} + 1\right ) + O\left (x^{6}\right ) \]