HP-35s SOLVE with integration

[Rick314]

I want to find the first null in Bessel function J0(x) using an HP-35s. The code below finds Jn(x). But trying to SOLVE it for J0(x) = 0 (at x = 2.4048) results in a "SOLVE ACTIVE" error at the "FN= I" instruction. I finally found in the manual under "Restrictions on Solving and Integrating" that "SOLVE and <integral>FN cannot call a routine that contains an FN=label instruction." This surprised me because older calculators such as the HP-15C can SOLVE an expression involving an integral. Is there any way to SOLVE J0(x) = 0 (at x = 2.4048) using an HP-35s?

Code:

LBL J  Jn(x) = (integral 0 to pi of LBL I)/pi
0      low lim
pi     hi lim
RAD
FN= I
<integrate>FN d T
pi
/
RTN

LBL I  COS(N*T - X*sin(T))
RCL N
RCL T
*
RCL X
RCL T
SIN
*
-
COS
RTN

J0(1): 0 STO N 1 STO X XEQ J ENTER -> 0.7652 (correct)