Question

What does it mean for a differential equation to be homogenous? I don't quite understand the notation in the definition.

Answer 1

I believe it basically means that if you add together a set of solutions, you still get a solution, and multiples of solutions are likewise solutions to the differential equation.

Hope someone else can expand further :)

Answer 2

So how can I recognize a homogenous equation?

Answer 3

If a differential equation is in terms of y (i.e. y'' + y' + y, etc.), it is said to be homogeneous as long as there is no term without a y. So for example

$$\frac{dy}{dx} + y = 0$$

is homogeneous, but

$$\frac{dy}{dx} + y = 13$$

is not, because the 13 is not in terms of the y.