show that the following set of functions are subspaces of F (-infinity, infinity)

a) all everywhere differentiable function that satisfy f'+2f=0 ... ?

Let suppose there are two functions that satisfy the equations

f ' + 2f = 0

and g' + 2g = 0

If we add both of them

f' + g' + 2f + 2g = 0

Grouping both the sides

(f+g) ' + 2 (f + g) = 0So, if both g and f satisfies it then their sum also satisfy it, So there is a subspace and their sum satisfy them too.