Section 5.4 Bessel’s Equation

(x^2)y\!\(\*
StyleBox["\"", "Subtitle"]\)\!\(\*
StyleBox[" ", "Subtitle"]\)\!\(\*
StyleBox["+", "Subtitle"]\)\!\(\*
StyleBox[" ", "Subtitle"]\)\!\(\*
StyleBox["xy", "Subtitle"]\)\!\(\*
StyleBox["'", "Subtitle"]\)\!\(\*
StyleBox["+", "Subtitle"]\)\!\(\*
StyleBox[" ", "Subtitle"]\)\!\(\*
StyleBox["(", "Subtitle",
FontFamily->"Gill Sans MT",
FontSize->24]\)\!\(\*
StyleBox[
SuperscriptBox["x", "2"], "Subtitle"]\)\!\(\*
StyleBox["-", "Subtitle"]\)\!\(\*
StyleBox[
SuperscriptBox["ν", "2"], "Subtitle"]\)\!\(\*
StyleBox[")", "Subtitle",
FontFamily->"Gill Sans MT",
FontSize->24]\)\!\(\*
StyleBox[" ", "Subtitle",
FontFamily->"Gill Sans MT",
FontSize->24]\)\!\(\*
StyleBox["y", "Subtitle"]\)\!\(\*
StyleBox[" ", "Subtitle"]\)\!\(\*
StyleBox["=", "Subtitle"]\)\!\(\*
StyleBox[" ", "Subtitle"]\)\!\(\*
StyleBox["0", "Subtitle"]\)

ν=0

y[x_]=.

eqn=x^2 y''[x]+ x y'[x]+x^2 y[x]==0

sol=DSolve[eqn,y[x],x]

eqn /. sol /. {C[1]-> 1, C[2] -> 0}

FullSimplify[%]

Plot[{BesselJ[0,x],BesselJ[1,x],BesselJ[2,x]},{x,0,20},PlotLegends->"Expressions"]

Gamma[1]

Gamma[2]

Gamma[3]

Gamma[4]

Plot[Gamma[x],{x,1,5}]