[Mathematics][MIT 18.03] Proof of a Theory about the Solution to Second-order Linear Homogeneous Differential Equation

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.

THM: if $y_{1},y_{2}$ are solu's to ODE, then either $W(y_{1},y_{2}) = 0$ (i.e. for all x) or$W(y_{1},y_{2}) is nonzero$ (i.e. for all x).

note: W means Wronskian

Well, frankly speaking, I got the inspiration from that introduction in wiki too.