**Solution:** Consider the non-linear partial differential equation f(x, y, z, p, q)= 0 —————————-(1)

Since z is a function of x and y, it follows that

dz = pdx + qdy —————————————–(2)

Let us assume p=u (x, y, z, a), where a is arbitrary constant, substitute in (1) solve to obtain q= v(x, y, z, a). For these value (2) becomes

dz = udx + vdy —————————————-(3)

Now if (3) can be integrated, yielding

g(x, y, z, a, b)=0———(4)

This is a complete solution of (1).

Since the success of the above procedure depends upon making a fortunate choice for p, it cannot be suggested as a standard procedure. We turn now to a general method for solving (1). This consists in finding an equation

F(x, y, z, p, q) = 0 ————–(5)

Such that (1) and (10) may be solved for p = P(x, y, z) and q= Q(x, y, z), that is, such that

and such that for these values of p and q the total differential equation.

*d*z = *p*dx + *q*dy = P(x, y, z)+Q(x, y, z) ——————————(7)

is integrable, that is

Differentiating (1) and (5) partially with respect to x and y, we find

Multiplying (8)by ∂F/∂p, (9) by ∂F/∂q, (10) by -∂f/∂p and (11) by -∂f/∂q and adding we obtain

This is a linear partial differential equation in F, considered as a function of the independent variables x, y, z, p, q. The auxiliary system is

Thus, we may take for (5) any solution of this system which involves p or, q, or both, which contains an arbitrary constant, and for which (6) holds.