OK, first off, clearly x(t) DOES equal 0 at a value of t, namely t = 0. So I assume you meant that it holds that x(t) doesn't equal 0 for all OTHER t.

You've gotten as far as

.

Using the initial condition

.

So

.

If then either or . The latter only occurs at , as was already stated by the initial condition.

So the only other way that is if .

Recall that as stated originally.

So .

This implies that .

Therefore, for any .