54.5.11 problem 11 (solved as direct Bessel)

Internal problem ID [8626]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 18. Power series solutions. 18.6. Indicial Equation with Equal Roots. Exercises page 373
Problem number : 11 (solved as direct Bessel)
Date solved : Sunday, March 30, 2025 at 01:21:38 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x y^{\prime \prime }+y^{\prime }-x y&=0 \end{align*}

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

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

\[ y = c_1 \operatorname {BesselI}\left (0, x\right )+c_2 \operatorname {BesselK}\left (0, x\right ) \]

✓ Mathematica. Time used: 0.017 (sec). Leaf size: 26

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

\[ y(x)\to c_1 \operatorname {BesselJ}(0,i x)+c_2 \operatorname {BesselY}(0,-i x) \]

✓ Sympy. Time used: 0.198 (sec). Leaf size: 14

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

\[ y{\left (x \right )} = C_{1} I_{0}\left (x\right ) + C_{2} Y_{0}\left (i x\right ) \]