45.3.16 problem 18

Internal problem ID [7274]
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 : 18
Date solved : Wednesday, March 05, 2025 at 04:22:28 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 x^{2} y^{\prime \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: 0.004 (sec). Leaf size: 36

Order:=6; 
ode:=4*x^2*diff(diff(y(x),x),x)+(16*x^2+1)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-x^{2}+\frac {1}{4} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (x^{2}-\frac {3}{8} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \right ) \sqrt {x} \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 69

ode=4*x^2*D[y[x],{x,2}]+(16*x^2+1)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \sqrt {x} \left (\frac {x^4}{4}-x^2+1\right )+c_2 \left (\sqrt {x} \left (x^2-\frac {3 x^4}{8}\right )+\sqrt {x} \left (\frac {x^4}{4}-x^2+1\right ) \log (x)\right ) \]

✓ Sympy. Time used: 0.870 (sec). Leaf size: 22

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*x**2*Derivative(y(x), (x, 2)) + (16*x**2 + 1)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

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