Existence and Uniqueness for Second Order ODE --Proof

The text we are using in class (Elementary Differential Equations by Boyce and DiPrima - 9th) provides a fairly complete yet not overly rigorous proof for the existence and uniqueness of solutions for first order ODE's using the Picard Lindelof theorem and the Lipschitz Condition. This is in Chapter 2.8 if anybody has this book.

The book then provides a uniqueness theorem for equations of the second order (Chapter 3.2) , however, it states that the proof is overly complex and it is thus omitted from the text. I find this rather disappointing as the uniqueness theorem for second order ODE's seems like it should be natural extension for that of first order ODE's.

I was wondering if anybody would be able to provide a link to a proof of this existence and uniqueness theorem... or if anybody cares to take the time to post it on this forum, it would make my day.

P.S. I do know that the existence and uniqueness for a second order equation can be proven by decomposing the equation into a system of first order ODE's and applying the Picard Lindelof method, (actually I feel you could do with with any n-th order ODE), however, I was wondering if there were any direct theorems.

Thanks,

Cheers