In what order should I teach methods for solving (linear) ODEs?

Question

I'm teaching at high school level, and the next topic will be differential equations. I would like my students to be able to solve some simple linear and non-linear ODEs using the ansatz $x(t) = e^{\lambda t}$, separating variables, varying the constant, etc. Also, I would like to treat both homogeneous and inhomogeneous linear differential equations. Lastly, I'd like to finish with some simple $2 \times 2$ or $3 \times 3$ systems of differential equations.

Many of these sections are strongly connected or can be applied to other sections, so I am having troubles choosing a good order in which I should teach the topics. The first order I've come up with is as follows:

1. Definitions and many examples
2. Direct integration
3. Separating variables
4. $x(t) = e^{\lambda t}$ ansatz
5. Principle of superposition
6. Varying the constant
7. Inhomogeneous linear ODEs
8. Systems in matrix notation and how to solve them

Question 1: Is there anything important that is missing?

Question 2: Is this order fine, or would you go for a (completely) different one? If so, why?

Answer 1

You left out linear first order ODE's, with their integrating factors. I use this often, compared to the other techniques. This probably should be done just after separation of variables, though I have seen it done before that topic.

I see why you would want to delay this topic until after you study inhomogeneous linear ODE's, for consistency. But the importance of this topic argues for an earlier look, even at the expense of this consistency. After all, you also put separation of variables early.

Answer 2

i think it would be nice for them to see the geometric idea of the integral curve through plotting the slope field. it works well for autonomous equations like the exponential growth and logistic model. i think it is important to do the first order equation really well and spend good amount of time in it; it pays dividend later when you are dealing with matrix equations.

here is an example. i first deal with $\frac{dx}{dt} = ax + f$ using undetermined coefficients and variation of parameters in finding particular solution. in fact the variation of parameter formula $x = \int_0^t e^{a(t-s)}f(s) \, ds$ readily extends to $\frac{dx}{dt} + Ax + f.$

the other system to spend time is the conservative system $\frac{d^2x}{dt^2} = -\frac{dV}{dx}.$ here the pendulum equation comes in handy.