54.7.9 problem 10 (as direct Bessel)

Internal problem ID [8657]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 18. Power series solutions. 18.9 Indicial Equation with Difference of Roots a Positive Integer: Logarithmic Case. Exercises page 384
Problem number : 10 (as direct Bessel)
Date solved : Sunday, March 30, 2025 at 01:22:37 PM
CAS classification : [_Bessel]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 15

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

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

✓ Mathematica. Time used: 0.018 (sec). Leaf size: 18

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

\[ y(x)\to c_1 \operatorname {BesselJ}(1,x)+c_2 \operatorname {BesselY}(1,x) \]

✓ Sympy. Time used: 0.213 (sec). Leaf size: 12

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

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