45.3.4 problem 4

Internal problem ID [7262]
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 : 4
Date solved : Wednesday, March 05, 2025 at 04:22:09 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 16 x^{2} y^{\prime \prime }+16 x y^{\prime }+\left (16 x^{2}-1\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 36

Order:=6; 
ode:=16*x^2*diff(diff(y(x),x),x)+16*x*diff(y(x),x)+(16*x^2-1)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_{2} \sqrt {x}\, \left (1-\frac {1}{5} x^{2}+\frac {1}{90} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{1} \left (1-\frac {1}{3} x^{2}+\frac {1}{42} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{{1}/{4}}} \]

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 52

ode=16*x^2*D[y[x],{x,2}]+16*x*D[y[x],x]+(16*x^2-1)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \sqrt [4]{x} \left (\frac {x^4}{90}-\frac {x^2}{5}+1\right )+\frac {c_2 \left (\frac {x^4}{42}-\frac {x^2}{3}+1\right )}{\sqrt [4]{x}} \]

✓ Sympy. Time used: 1.037 (sec). Leaf size: 42

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(16*x**2*Derivative(y(x), (x, 2)) + 16*x*Derivative(y(x), x) + (16*x**2 - 1)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \sqrt [4]{x} \left (\frac {x^{4}}{90} - \frac {x^{2}}{5} + 1\right ) + \frac {C_{1} \left (\frac {x^{4}}{42} - \frac {x^{2}}{3} + 1\right )}{\sqrt [4]{x}} + O\left (x^{6}\right ) \]