Is it possible to write a vector form ODE to an explicit form?

Question

Any explicit differential equation of order n, $F\left(x,y,y',y'',\ \ldots ,\ y^{(n-1)}\right)=y^{(n)}$ can be written as a system of n first-order differential equations.

Conversely, can any vector form be rewritten into an explicit form?

Answer 1

The answer is not.

• With a explicit form always we can re-write it as a implicit form.
• With a implicit form not always we can re-write it as a explicit form.

An example for the second case is a Clairaut ordinary differential equation $$y=xy'+(y')^{2}$$ Also, you can find more example in implicit differential equation.