[llamaz]

I think the calculus of variations might be a better approach to introducing ODEs in first year.

You can show that by generalizing calculus so the values are functions rather than real numbers, then trying to find a max/min using the functional version of dy/dx = 0, you end up with an ODE (viz. the Euler-Lagrange equation).

This also motivates Lagrange multipliers which are usually taught around the same time as ODEs. They are similar to the Hamiltonian, which is a synonym for energy and is derived from the Euler-Lagrange equations of a system.

Of course you would brush over most of this mechanics stuff in a single lecture (60 min). But now you've motivated ODEs and given the students are reason to solve ODEs with constant coefficients.

[whatshisface]

You don't need a Lagrangian to invent mechanical ODEs. You could talk about mixing tanks, objects under complicated forces, and so on with a lot less background information.