# Finding Coefficients involving Bessel Functions

I have a differential equation: $$x^2 \frac{d^2 y}{dx^2} +(x^2-3)y=0$$

I solved it and obtained:

y[x] -> Sqrt[x] BesselJ[Sqrt[13]/2, x] C[1] + Sqrt[x] BesselY[Sqrt[13]/2, x] C[2]

Now I'm trying to find the value for C[2] and C[1] when y[x]=0 at x=0 and 1 at x=1.

I tried using simultaneous equations using the Solve[ ] function such as:

eq1 = Sqrt[x] BesselJ[Sqrt[13]/2, x] C[1] = 0
eq2 = Sqrt[x] BesselJ[Sqrt[13]/2, x] C[1] = 1
Solve[{eq1, eq2}, C[1]]


It appears that x=0 is a singular point for this. We can go about it less directly.

yx =
DSolveValue[{x^2*D[y[x], {x, 2}] + (x^2 - 3)*y[x] == 0, y[1] == 1},
y[x], x]

(* Out[524]= 1/
BesselY[Sqrt[13]/2,
1] (Sqrt[x] BesselY[Sqrt[13]/2, x] +
Sqrt[x] BesselJ[Sqrt[13]/2, x] BesselY[Sqrt[13]/2, 1] C[1] -
Sqrt[x] BesselJ[Sqrt[13]/2, 1] BesselY[Sqrt[13]/2, x] C[1]) *)


(Something like the above, maybe without the IC, should have been in the original post by the way. I really do not like to do unnecessary typing.)

So now we have one free parameter. We can get a general sort of solution for the case where y[x]==x, then extract an appropriate limit. First solve for when the expression in x is equal to x.

csmall = C[1] /. First[Solve[yx == x, C[1]]]

(* Out[543]= (Sqrt[x] BesselY[Sqrt[13]/2, 1] - BesselY[Sqrt[13]/2, x])/(
BesselJ[Sqrt[13]/2, x] BesselY[Sqrt[13]/2, 1] -
BesselJ[Sqrt[13]/2, 1] BesselY[Sqrt[13]/2, x]) *)


Now see if we can evaluate for x==0. Just plugging in will give an indeterminate form, so I take a limit instead.

c0 = Limit[csmall, x -> 0]

(* Out[544]= 1/BesselJ[Sqrt[13]/2, 1] *)


Now use that for the remaining constant.

yxComplete = yx /. C[1] -> c0

(* Out[547]= (Sqrt[x] BesselJ[Sqrt[13]/2, x])/BesselJ[Sqrt[13]/2, 1] *)

• What about a value for C[2]? Thanks for the answer by the way – Ray Jan 19 '20 at 1:17
• There is no C[2] (check Out[524]). I used one boundary condition so there was only one undetermined parameter. – Daniel Lichtblau Jan 19 '20 at 15:54