# Elementary Proof Involving Wronskian and Existence and Uniqueness

I’m in need of some help to understand an elementary proof involving the Wronskian of $$n$$ vector functions and the following existence and uniqueness theorem.

Existence and Uniqueness

Suppose $$A(t)$$ and $$f(t)$$ are continuous on an open interval $$I$$ that contains the point $$t_0$$. Then, for any choice of initial vector $$x_0$$, there exists a unique solution $$x(t)$$ on the whole interval $$I$$ to the initial value problem

$$x’(t) = A(t)x(t) + f(t) \space \space , \space \space x(t_0) = x_0$$

The proof I’m reading is about how if $$x_1, \space \ldots \space, x_n$$ are linearly independent solutions on $$I$$ then the Wronskian is never zero. So I understand that if the Wronskian is zero this implies that the vector functions are linearly dependent which implies that $$c_1 x_1(t_0) \space + \space \ldots \space + \space c_n x_n(t_0) = 0$$

At some point $$t_0$$ (ie a non-trivial solution exists at some point). Now here is what my textbook goes on to say:

… However, $$c_1 x_1(t_0) \space + \space \ldots \space + \space c_n x_n(t_0) = 0$$ and the vector function $$z(t) \equiv 0$$ are both solutions to $$x’ = Ax$$ on I, and they agree at the point $$t_0$$. So these solutions must be identical on $$I$$ according to the existence and uniqueness theorem (What!?).

So this may be obvious to some, but I’m having a hard time understanding what the existence and uniqueness theorem means in this context and how it proves that it is zero for all $$t$$. Any explanation would be appreciated.

You have two solutions, one is the linear combination $\sum_i c_ix_i$ the other is the nulfunction $z\equiv 0$. Both of these solutions are equal to $0$ for the point $t_0$.
Uniqueness of the solution then tells you that both have to be the same otherwise there would be two different solutions satisfying the same differential equation and initial condition in $t_0$.