At first, I'd like to say thank you to MIT open courses which give me the privilege to enjoy the most outstanding education resources.
Okay, come to the point. When I was learning the second-order homogeneous differential equation, the professor quoted a theory in one step to prove that ${c_{1}y_{1}+c_{2}y_{2}}$ are all the solutions.
Well, frankly speaking, I got the inspiration from that introduction in wiki too.