Name of theorem which says there are n solutions to an nth order homogeneous ODE Not a very thought-provoking question to get me started but here it goes; I am writing an undergraduate report on solving a homogeneous ODE. The ODE in question is a second-order ODE, so it must have two linearly-independent solutions (right?). I know this myself but I want to justify it in my report by saying something like "it is known that there are two linearly-independent solutions to this ODE from Johnny's theorem", obviously subbing in the actual name of the theorem for "Johnny's theorem". Perhaps the statement of Johnny's theorem would be something like "there are n linearly-independent solutions to an nth order ODE". It seems like a pretty important theorem that would have its own name. Does anyone know the name of this theorem (if possible give me a link to a Wikipedia/other page detailing it)? I don't need to prove it, I just want to refer to it in my report. Thanks.
 A: I'm not sure there is a particular name that sums everything up, but I think there are two important concepts, and the first does have a name.
The first concept is linearity. For a linear operator $L$, $L(y_0+y_1)$ = $Ly_0 + Ly_1$, so if you have $y_0$ and $y_1$ that individually satisfy $Ly=0$ then $y_0+y_1$ must also satisfy $Ly=0$. This is known as the Superposition Principle. It applies to linear systems generally but it has particular application to differential equations, where $L=p_2(x)\frac{d}{dx^2} + p_1(x)\frac{d}{dx} + p_0(x)$.
The second concept is that the most general solution to an $n$th order differential equation has $n$ arbitrary concepts. I don't think this has a name: I've always seen it written out long-hand.
Putting these together then a weighted sum of two linearly independent solutions gives you the most general solution to a second order linear homogeneous differential equation.
A: Fundamental matrix, fundamental basis and Wronski determinant are concepts that come to mind. The theorem about the Wronskian essentially tells you that solutions that are linearly independent at one point stay independent for all times. Linear independence is defined for the first order system connected to the higher order equation.
