Question

Form the partial differential equation by the elimination of arbitrary constants and function from log(az-1)=x+ ay+ b

a) ∂^2z/∂x^2 = a
b) ∂^2z/∂y^2 = ax + b
c) ∂z/∂x = a
d) ∂z/∂y = ax + b

Answer 1

By differentiating the given equation with respect to 'x' and then 'y', we eliminate the arbitrary constants 'a' and 'b', resulting in a partial differential equation (∂^2z/∂x∂y = 0) where neither 'a' nor 'b' appears.

Step-by-step explanation:

The task is to form a partial differential equation by eliminating arbitrary constants 'a' and 'b' from the equation log(az-1) = x + ay + b. Firstly, we differentiate the equation with respect to 'x' to get ∂z/∂x, which would eliminate 'b'. Next, we differentiate the resulting equation with respect to 'y' to eliminate 'a'.

The first derivative with respect to 'x' gives us ∂z/∂x = a/(az-1). Differentiating again with respect to 'y', we obtain ∂^2z/∂x∂y = 0, because the partial derivative of 'a' with respect to 'y' is zero (as 'a' is a constant not depending on 'y'). Since there's no term involving 'x' in the double derivative, options (b) and (d) are incorrect.