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I have come across a PDE of the following form in my research:

C_1 \alpha(t,r) + C_2 \partial_t \alpha(t,r) + C_3 \partial_r \alpha(t,r) + C_4 \beta(t,r) + C_2 \partial_t \beta(t,r) + C_3 \partial_r \beta(t,r) = 0

where the coefficients C_i are all functions of t and r: C_i = C_i(t,r). I want to solve for the functions \alpha(t,r) \beta(t,r). I understand that this is one PDE for two unknown functions. I guess what I would like to do is solve for \alpha(t,r) in terms of \beta(t,r).

In addition, I would also like to solve it subject to the constraint

A_1 \alpha(t,r) + A_2 \partial_t \alpha(t,r) + A_3 \beta(t,r) + A_4 \partial_r \beta(t,r) = 0

where once again the coefficients A_i are all functions of t and r: A_i = A_i(t,r).

Any suggestions or names of methods used to solve equations or I guess really a system of equations when the constraint is considered would be helpful. Thanks in advanced!!