Method of seperation of variables

A function has input scalars/vectors, and output scalars/vectors. Similarly operators input functions and output functions.

An operator is linear if and only if it is satisifed that

L(c_1u_1 + c_2u_2) = c_1L(u_1) + c_2L(u_2)

Where u_1,u_2 are arbitrary functions and c_1,c_2 are arbitrary constants.

The heat operator is

L(u) = \frac{\partial u}{\partial t} - k\frac{\partial^2 u}{\partial x^2}

A linear equation for the unknown u is of the format

L(u) = f(x,t)

Where L is a linear operator and f(x,t) does not depend on u.

Moreover if in the linear equation L(u) = f(x,t) we have f(x,t) = 0 we say the equation is Linear and homogeneous.