Why is the Nth derivate isolated in many textbook definitions? 
My textbook keeps writing definitions in the above form, with the LHS of the equation being the highest order derivative, and the RHS being everything else. Why is this form stressed over a form like this

? Is there any mathematical reason, if it is just a convention then why?
Notice that the second equation has a function of t on the RHS and everything else on the LHS.
 A: As pointed out in a comment, the $\phi^{(n)}(t)$ term has to be there, or else it wouldn't be an $n$th order differential equation, but all of the other stuff could be zero.
I think there is one other reason that it is often useful to write things this way.  It is more conducive to numerical solution of the differential equation.  The simplest way to solve a differential equation numerically is the Euler method, where you basically take the initial conditions for $\phi$ and all its lower derivatives and use this to calculate $\phi^{(n)}(t_0)$.  Then you use this to estimate the function and all of its lower derivatives at a nearby point $t_1$, and then you once again have to calculate $\phi^{(n)}(t_1)$ based on the other information.  The only nontrivial step is calculating $\phi^{(n)}(t)$, so it is nice to have the equation in a form where that is easy to calculate.
A: Because in this explicit form, if the right side function $f$ is continuous (on some open domain) then you can directly say that this is an ODE. Then all the basic existence and uniqueness theorems can be applied without further restrictions.
In contrast, if the equation is in implicit form, and if $y$ is a vector, then it is not given that one can isolate the highest derivatives (or even that the highest order derivatives have the same order for all variables), one would have to demand the conditions of the implicit function theorem. Even then, there can be multiple solutions for the highest derivative. See Clairaut equations for simple examples.
Without that local solvability for the highest derivatives, an implicit differential equation could as well be a DAE, a differential-algebraic system of equations (or in exotic cases fail to fall into any classification).
