The Wronskian and the term "fundamental set of solutions"
Let us given a second order homogenous linear differential equation and I'm given an initial value problem. Suppose I found two solutions satisfying this differential equation and the initial values.
By Theorem 1 , If their Wronskian at the initial value is not zero at $t_0$ I know that their linear combination is a solution too.
I'm also aware of Theorem 2 that, if Wronskian of these two solutions are nonzero at a point $t$ then their linear combination is the general solution.
So doesn't Theorem 1 imply Theorem 2 ?
By the way what is meant by "fundamental set of solutions"? I am given the following theorem 3 :
Theorem 3 states that these two solutions of the DE are said to form a fundamental set of solutions if and only if their Wronskian is nonzero.
From "Wronskian in nonzero", I understand that Wronskian shouldn't be zero at any $t$.But may I ask, what is "fundamental set of solutions"? I believe it is some sort of a stronger relationship between two solutions than the general solution due to Wronskian being nonzero at any $t$. So how are "fundamental set of solutions" helpful to us and what is the difference between them and the "general solution"
$\endgroup$1 Answer
$\begingroup$A fundamental set of solutions to a differential equation is the basis of the solution space of the differential equation. Put in another way, every solution to a differential equation can be written as a linear combination of these fundamental solutions.
Secondly, the Wronskian being non-zero at a point $t$, tells us that the two solutions are linearly independent at that point. For two solutions to be the part of the basis for a solution space, we require them to be linearly independent.
Lastly, since the differential equation you are working with is of second order, the fundamental solution set consists of two linearly independent solutions. These two linearly independent solutions span the solution space (and hence form the basis for the solution space).
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