23.3.289 problem 291

Internal problem ID [6003]
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 : 291
Date solved : Tuesday, September 30, 2025 at 02:19:41 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.009 (sec). Leaf size: 27

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

\[ y = \frac {c_1 \operatorname {BesselJ}\left (n +\frac {1}{2}, a x \right )+c_2 \operatorname {BesselY}\left (n +\frac {1}{2}, a x \right )}{\sqrt {x}} \]

✓ Mathematica. Time used: 0.017 (sec). Leaf size: 22

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

\begin{align*} y(x)&\to c_1 j_n(a x)+c_2 y_n(a x) \end{align*}

✓ Sympy. Time used: 0.203 (sec). Leaf size: 41

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

\[ y{\left (x \right )} = \frac {C_{1} J_{\sqrt {n \left (n + 1\right ) + \frac {1}{4}}}\left (a x\right ) + C_{2} Y_{\sqrt {n \left (n + 1\right ) + \frac {1}{4}}}\left (a x\right )}{\sqrt {x}} \]