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**2nd Order Homogenous ODE (Two solutions??)**

Alright.

I understand that if we have a differential equation of the form

[itex]A\cdot\frac{d^{2}y}{dt}+B\cdot\frac{dy}{dt}+C\cdot y = 0[/itex]

and it has the solution y

_{1}(t), and y

_{2}is also a solution. Then any combination of the two

y

_{H}=C

_{1}y

_{1}(t)+C

_{2}y

_{2}(t) is also a solution.

But, mathmatically speaking, so would a combination with a third "possible" solution y

_{3}.

Now I know there must be some theorem stating that there will only be two solutions for this type of ODE, but can anyone tell me where I can find these?

ALSO,

If there are two solutions, why do we always use the combination of the two? Why not just pick one of the solutions and use it? Why "overcomplicate" it and use the combination of the two solutions as the general solution?

I know there must be a good reason, but I havent found it, and I need someone to point it out to me, or tell me where I can read about it.

Thank you in advance,

Rune