Application of Orthogonality condition

Application of Orthogonality condition

I have applied separation of variables to a transient radial heat equation
problem.
T is a function of r and t.
I have reached the following step:
$T_2(r,t) = \sum_{m=1}^ \infty c_m
e^{-\alpha_2\lambda_m^2t}\left(\dfrac{-Y_0(\lambda_mb)J_0(\lambda_m
r)}{J_0(\lambda_mb)}+Y_0(\lambda_mr)\right)$
I need to find $c_m$ which is usually found using orthogonality condition.
This is done by multiplying both sides with $rJ_0(ë_nr)$ and integrating
both sides with respect to r. But here in above equation I can't apply the
orthogonality condition due to presence of $Y_0(ë_mr)$ on right side of
equation.
How do I apply orthogonality condition in this case?
I just want to find out the coefficient $c_m$ and somehow get rid of the
summation sign.
Please help!