52.3.16 problem 18

Internal problem ID [8302]
Book : DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL, WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th edition.
Section : CHAPTER 6 SERIES SOLUTIONS OF LINEAR EQUATIONS. 6.4 SPECIAL FUNCTIONS. EXERCISES 6.4. Page 267
Problem number : 18
Date solved : Wednesday, March 05, 2025 at 05:34:31 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*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 23

ode:=4*x^2*diff(diff(y(x),x),x)+(16*x^2+1)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left (\operatorname {BesselJ}\left (0, 2 x \right ) c_{1} +\operatorname {BesselY}\left (0, 2 x \right ) c_{2} \right ) \sqrt {x} \]

✓ Mathematica. Time used: 0.021 (sec). Leaf size: 28

ode=4*x^2*D[y[x],{x,2}]+(16*x^2+1)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \sqrt {x} (c_1 \operatorname {BesselJ}(0,2 x)+c_2 \operatorname {BesselY}(0,2 x)) \]

✓ Sympy. Time used: 0.091 (sec). Leaf size: 20

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)
 

\[ y{\left (x \right )} = \sqrt {x} \left (C_{1} J_{0}\left (2 x\right ) + C_{2} Y_{0}\left (2 x\right )\right ) \]