Final answer:
To find the PDE for which the given solution z = xf(x/y) + yg(x/y) applies, we set u = x/y and use the chain rule on z to eliminate the functions f and g by finding a relation between the first and second order partial derivatives of z concerning x and y.
Step-by-step explanation:
The student question involves finding a partial differential equation (PDE) for which the given general solution z = xf(x/y) + yg(x/y) applies. Since the general solution involves arbitrary functions f and g of x/y, we need to differentiate to eliminate these functions and find a PDE that does not contain them.
To start, let's denote u = x/y. Now recognizing that f(u) and g(u) are functions of a single variable u, we apply the chain rule for partial differentiation to z. First, we calculate the first-order partial derivatives of z concerning x and y and then the second-order partial derivatives. By setting up these derivatives, we aim to find an expression that will eliminate the arbitrary functions f and g to get a relation solely in terms of z, x, and y.
Applying this procedure, the sought PDE that eliminates the arbitrary functions f and g from the general solution will likely involve a relationship between first and second-order partial derivatives of z concerning x and y.