45.3.18 problem 20

Internal problem ID [7276]
Book : A FIRST COURSE IN DIFFERENTIAL EQUATIONS with Modeling Applications. Dennis G. Zill. 9th edition. Brooks/Cole. CA, USA.
Section : Chapter 6. SERIES SOLUTIONS OF LINEAR EQUATIONS. Exercises. 6.3.1 page 250
Problem number : 20
Date solved : Sunday, March 30, 2025 at 11:53:12 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 9 x^{2} y^{\prime \prime }+9 x y^{\prime }+\left (x^{6}-36\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.024 (sec). Leaf size: 27

Order:=6; 
ode:=9*x^2*diff(diff(y(x),x),x)+9*x*diff(y(x),x)+(x^6-36)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \,x^{2} \left (1+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_2 \left (-144+\operatorname {O}\left (x^{6}\right )\right )}{x^{2}} \]

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 16

ode=9*x^2*D[y[x],{x,2}]+9*x*D[y[x],x]+(x^6-36)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_2 x^2+\frac {c_1}{x^2} \]

✓ Sympy. Time used: 1.046 (sec). Leaf size: 22

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(9*x**2*Derivative(y(x), (x, 2)) + 9*x*Derivative(y(x), x) + (x**6 - 36)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = \frac {C_{2} \left (1 - \frac {x^{6}}{108}\right )}{x^{2}} + C_{1} x^{2} + O\left (x^{6}\right ) \]