85.48.2 problem 2 (b)

Internal problem ID [22798]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter 4. Linear differential equations. C Exercises at page 190
Problem number : 2 (b)
Date solved : Thursday, October 02, 2025 at 09:14:42 PM
CAS classification : [_Bessel]

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

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

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

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

✓ Mathematica. Time used: 0.048 (sec). Leaf size: 18

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

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

✓ Sympy. Time used: 0.155 (sec). Leaf size: 22

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

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