Let’s consider the function F(K,L) and the following system of partial differential equations:

∂lnF/∂K = (1-α)/K (1)

∂lnF/∂L = α/L (2)

I have a solution for the above system of differential equations but am missing some of the details of its development.

STEP 1

First, the solution says that based on the fact that ∫1/x= ln(x) +c (where c is a constant of integration), we can write:

ln F(K,L) = (1 - α) lnK + g(L) + c (1’)

ln F(K,L) = α lnL + h(K) + c’ (2’)

where g(L) and h(K) are constants of integration that may depend on L and K respectively.

c and c’ are also constants of integration that do not depend on any of the two indicated variables L and K.

STEP 2

Next , (1’) and (2’) are combined to get to the following expression:

ln F(K,L) = (1 - α) lnK + α lnL + C (3)

¿can anyone give me more theoretical background and details to understand STEP 1 and STEP 2? That's to say: how to use integrals to solve the system of differential equations and how to combine (1') and (2') to get to (3)

The context of the equations is the field of economics (F is a production function, K is capital, L is labor and α is the fraction paid to labor. In fact, I am mostly interested in getting support to understand the mathematical side of the posted issue .

Thanks.