23.3.446 problem 451

Internal problem ID [6160]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 3. THE DIFFERENTIAL EQUATION IS LINEAR AND OF SECOND ORDER, page 311
Problem number : 451
Date solved : Tuesday, September 30, 2025 at 02:23:54 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (4 a^{2} x^{2}+1\right ) y+4 x^{2} y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 23

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

\[ y = \left (\operatorname {BesselJ}\left (0, x a \right ) c_1 +\operatorname {BesselY}\left (0, x a \right ) c_2 \right ) \sqrt {x} \]

✓ Mathematica. Time used: 0.014 (sec). Leaf size: 28

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

\begin{align*} y(x)&\to \sqrt {x} (c_1 \operatorname {BesselJ}(0,a x)+c_2 \operatorname {BesselY}(0,a x)) \end{align*}

✓ Sympy. Time used: 0.049 (sec). Leaf size: 20

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(4*x**2*Derivative(y(x), (x, 2)) + (4*a**2*x**2 + 1)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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