Let Sc be the set of differentiate functions ux that satisfy

Let S_c be the set of differentiate functions u(x) that satisfy the differential equation u\'(x) = 2xu(x) + c for all real x. For which valuer(s) of the real constant c is this set a linear subspace? Prove.

Solution

u\'(x) = 2xu(x) + C

Let u(x) = 0 then u\'(x) = 0

Hence 0=0+C => C = 0

Now for u,v in S

Consider, u+v

Then u\'(x) + v\'(x) = 2x(u(x) + v(x) + C

=> (u+v)\' (x) = 2xu(x)+2xv(x)+C

=> (u+v)\'(x) = (u\'-C) + (v\'-C) + C

=> (u+v)\'(x) = u\'+v\'-C

And hence C has to be 0

So for S to be a subspace , C must be equal to 0