This is the basic definition of integral when you calculate integral product of orthogonal Bessel functions.
What happens when you change integral bounds from [0,a] to [b,c]? Do you lose the orthogonality or does it remain? What is the solution of the same integral only with boundaries [b,c]
P.S.: This is my initial equation.
Cs and Ds are unknown constants, but i can express Ds with using Cs - Cs is the constant i am looking for. Xs and sn are determined numerically i can easily determine them, not the point. The plan (according to what we did in school) is that i multiply this with orthogonal function and integrate it so only m=n product give non-zero integrals.
How am i supposed to solve this problem if the functions are not orthogonal? How can i find Cs?
P.S. P.S.: This is the correlation between Cs and Ds.
Take $a=1$. Consider the first two zeros of $J_0$; call then $a_{01}$ and $a_{02}$. Then $$ \int_0^T J_0(a_{01}\rho)J_0(a_{02}\rho)\rho\;d\rho = \frac{a_{01}J_1( a_{01}T)J_0(a_{02}T) - a_{02}J_0(a_{01}T)J_1(a_{02}T)}{(a_{01}^2 - a_{02}^2)}\;T $$ If we put $T=1$, then both numerator terms vanish, because we get $J_0(a_{01})=0$ and $J_0(a_{02})=0$. If we use some other value of $T$, we do not get these zero terms.