Question

How to detect that wether the equation is homogenous or linear?

Answer 1

we say that it is homogenous if and only if g ( x ) ≡ 0 . You can write down many examples of linear differential equations to check if they are homogenous or not. For example, y ″ sin ⁡ x + y cos ⁡ x = y ′ is homogenous, but y ″ sin ⁡ x + y tan ⁡ x + x = 0 is not and so on.