What is meant by homogeneous boundary conditions?

I am sorry if this is basic knowledge for differential equations but it has been a long time since I took the class, I probably learnt it and forgot about it. I would appreciate the explanation. Thank you!

Solution 1:

The simplest way to test whether an equation (here the equation for the boundary conditions) is homogeneous is to substitute the zero function and see whether it equals to zero.

Solution 2:

If your differential equation is homogeneous (it is equal to zero and not some function), for instance, $$\frac{d^2y}{dx^2}+4y=0$$ and you were asked to solve the equation given the boundary conditions, $$y(x = 0) = 0$$ $$y(x = 2\pi) = 0$$

Then the boundary conditions above are known as homogenous boundary conditions.

It is important to remember that when we say homogeneous (or inhomogeneous) we are saying something not only about the differential equation itself but also about the boundary conditions as well.

Formulation and motivation for this answer have been summarized from Paul's Online Notes