Question

Is the set of all solutions to the differential equation y'+9y = 4x^2 closed under addition and closed under scalar multiplication? ( do not solve the ODE)
Please, explain me

Answer 1

Well, closed (closure) basically means that if you perform an operation, does your answer lie in the same set as the original components. For example, integers are closed under addition, subtraction and multiplication, but not under division. Because +, -, * will always result in another integrer, but division may not. So basically, if the ODE is, lets say, in the set of all real numbers, then of course it would be closed under addition and multiplication because when you add and multiply two real numbers, you always get two real numbers. So I suppose your job is to classify the set in which your ODE falls into and then state that if you add or multiply to it, do you still have the same set.

Answer 2

I got you, so it is because the ODE is a nonhomogeneous one whose RHS is a polynomial degree 2. That's the proof of set solution is P_2 and P_2 is always a vector space.

Answer 3

Pretty much, lol. Long as that makes sense :3