(ODE) I want to make sure I understand the general premise behind the conditions for a unique IVP solution existing; this isn't a question about a problem in and of itself, but about IVP's in general.

Question

mendicant_bias · Answer

I have a theorem in my book that states this: http://i.imgur.com/i4tCrpA.png

mendicant_bias · Answer

What I get from this is that the lead coefficient can never be zero in any instance that you plug in the argument of the initial conditions, e.g, let's say I had some IVP like

mendicant_bias · Answer

$$12y'''+3y''-9y'+30y=5x,$$Where your lead coefficient, a_n, is equal to 12, all of your coefficients are continuous, and your lead coefficient is nonzero for every x in the initial condition argument, because your lead coefficient is *independent* of x.

mendicant_bias · Answer

Let's say I had the general conditions, $$y(0) = 22, \ \ \ y(5)=39, \ \ \ y(77)=15;$$
What this theorem is saying, really, is that, apart from the straightforward continuous and linear stuff, for whatever argument of my initial conditions, the lead coefficient must *not* be zero, right?

mendicant_bias · Answer

Like, if I had a lead coefficient $$a_n(x)$$ for this problem, $$a_n(0), \ a_n(5), \ a_n(77) \neq 0$$in order for the theorem to be fulfilled and for some sort of unique solution to exist. @eliassaab , is this right?