39.3.3 problem 3

Internal problem ID [8547]
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 : 3
Date solved : Tuesday, September 30, 2025 at 05:38:56 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 x^{2} y^{\prime \prime }+4 x y^{\prime }+\left (4 x^{2}-25\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.023 (sec). Leaf size: 36

Order:=6; 
ode:=4*x^2*diff(diff(y(x),x),x)+4*x*diff(y(x),x)+(4*x^2-25)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_1 \,x^{5} \left (1-\frac {1}{14} x^{2}+\frac {1}{504} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (2880+480 x^{2}+120 x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{{5}/{2}}} \]

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 58

ode=4*x^2*D[y[x],{x,2}]+4*x*D[y[x],x]+(4*x^2-25)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (\frac {x^{3/2}}{24}+\frac {1}{x^{5/2}}+\frac {1}{6 \sqrt {x}}\right )+c_2 \left (\frac {x^{13/2}}{504}-\frac {x^{9/2}}{14}+x^{5/2}\right ) \]

✓ Sympy. Time used: 0.335 (sec). Leaf size: 42

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*x**2*Derivative(y(x), (x, 2)) + 4*x*Derivative(y(x), x) + (4*x**2 - 25)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} x^{\frac {5}{2}} \left (1 - \frac {x^{2}}{14}\right ) + \frac {C_{1} \left (- \frac {x^{6}}{144} + \frac {x^{4}}{24} + \frac {x^{2}}{6} + 1\right )}{x^{\frac {5}{2}}} + O\left (x^{6}\right ) \]