45.3.23 problem 25

Internal problem ID [7281]
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 : 25
Date solved : Sunday, March 30, 2025 at 11:53:19 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 16 x^{2} y^{\prime \prime }+32 x y^{\prime }+\left (x^{4}-12\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.025 (sec). Leaf size: 32

Order:=6; 
ode:=16*x^2*diff(diff(y(x),x),x)+32*x*diff(y(x),x)+(x^4-12)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_1 \,x^{2} \left (1-\frac {1}{384} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (-2+\frac {1}{64} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{{3}/{2}}} \]

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 40

ode=16*x^2*D[y[x],{x,2}]+32*x*D[y[x],x]+(x^4-12)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (\frac {1}{x^{3/2}}-\frac {x^{5/2}}{128}\right )+c_2 \left (\sqrt {x}-\frac {x^{9/2}}{384}\right ) \]

✓ Sympy. Time used: 1.054 (sec). Leaf size: 32

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(16*x**2*Derivative(y(x), (x, 2)) + 32*x*Derivative(y(x), x) + (x**4 - 12)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \sqrt {x} \left (1 - \frac {x^{4}}{384}\right ) + \frac {C_{1} \left (1 - \frac {x^{4}}{128}\right )}{x^{\frac {3}{2}}} + O\left (x^{6}\right ) \]