23.3.447 problem 452

Internal problem ID [6161]
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 : 452
Date solved : Tuesday, September 30, 2025 at 02:23:54 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 19

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

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

✓ Mathematica. Time used: 0.034 (sec). Leaf size: 38

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

\begin{align*} y(x)&\to c_1 \operatorname {Gamma}(1-a) \operatorname {BesselJ}\left (-a,\sqrt {x}\right )+c_2 \operatorname {Gamma}(a+1) \operatorname {BesselJ}\left (a,\sqrt {x}\right ) \end{align*}

✓ Sympy. Time used: 0.138 (sec). Leaf size: 29

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

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