39.3.8 problem 8

Internal problem ID [8552]
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 : 8
Date solved : Tuesday, September 30, 2025 at 05:39:01 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+x y^{\prime }+\left (36 x^{2}-\frac {1}{4}\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.023 (sec). Leaf size: 34

Order:=6; 
ode:=x^2*diff(diff(y(x),x),x)+x*diff(y(x),x)+(36*x^2-1/4)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_1 x \left (1-6 x^{2}+\frac {54}{5} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (1-18 x^{2}+54 x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{\sqrt {x}} \]

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 52

ode=x^2*D[y[x],{x,2}]+x*D[y[x],x]+(36*x^2-1/4)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (54 x^{7/2}-18 x^{3/2}+\frac {1}{\sqrt {x}}\right )+c_2 \left (\frac {54 x^{9/2}}{5}-6 x^{5/2}+\sqrt {x}\right ) \]

✓ Sympy. Time used: 0.335 (sec). Leaf size: 44

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) + x*Derivative(y(x), x) + (36*x**2 - 1/4)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \sqrt {x} \left (\frac {54 x^{4}}{5} - 6 x^{2} + 1\right ) + \frac {C_{1} \left (54 x^{4} - 18 x^{2} + 1\right )}{\sqrt {x}} + O\left (x^{6}\right ) \]