If is a solution to a first-order differential equation in the form of , then , with c-constant, is a solution as well.
I know the idea of the proof, but I am having trouble expressing it in a rigorous proof. Namely, my idea is that if we shift to the left by , then both the LHS and RHS will shift by (invoking the chain rule), so they are equal. Should I define and do a change of variable?