Question

For a partial differential equation, in a function φ (x, y) and two variables x, y, what is the form obtained after separation of variables is applied?

(a) Φ (x, y) = X(x)+Y(y)

(b) Φ (x, y) = X(x)-Y(y)

(c) Φ (x, y) = X(x)Y(y)

(d) Φ (x, y) = X(x)/Y(y)

The question was posed to me in final exam.

I want to ask this question from Method of Separation of Variables in chapter Applications of Partial Differential Equations of Engineering Mathematics

Answer 1

The correct answer is (c) Φ (x, y) = X(x)Y(y)

For explanation I would say: The method of separation of variables relies upon the assumption that a function of the form,

Φ (x, y) = X(x)Y(y)

will be a solution to a linear homogeneous partial differential equation in x and y. This is called a product solution and provided the boundary conditions are also linear and homogeneous this will also satisfy the boundary conditions.