how to solve a set of linear partial differential equations

Hi,

I just wonder if there are some analytical methods of solving partial
differential equations like the following:

diff(f(X),x_i) = g_i(X), for all i=1,...,N

where functions f,g_1,...,g_N are mappings from vector X=(x_1,...,x_N)
to a scalar.

Especially, we have

g_i(X) = x_i*(A-B*x_i) *
(1-x_1)*...*(1-x_{i-1})*(1-x_{i+1})*...(1-x_N), for all i.

where A and B are just constants.

Thanks!

.

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