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.

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