23.3.272 problem 274

Internal problem ID [5986]
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 : 274
Date solved : Tuesday, September 30, 2025 at 02:07:21 PM
CAS classification : [_Bessel]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 15

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

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

✓ Mathematica. Time used: 0.032 (sec). Leaf size: 18

ode=-((p^2 - x^2)*y[x]) + 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 \operatorname {BesselJ}(p,x)+c_2 \operatorname {BesselY}(p,x) \end{align*}

✓ Sympy. Time used: 0.134 (sec). Leaf size: 22

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

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