23.3.449 problem 454

Internal problem ID [6163]
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 : 454
Date solved : Tuesday, September 30, 2025 at 02:23:56 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 19

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

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

✓ Mathematica. Time used: 0.127 (sec). Leaf size: 26

ode=-(((1 + 2*n)^2 - 4*x^2)*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 {BesselJ}\left (n+\frac {1}{2},x\right )+c_2 \operatorname {BesselY}\left (n+\frac {1}{2},x\right ) \end{align*}

✓ Sympy. Time used: 0.170 (sec). Leaf size: 32

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

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