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

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?

• You might consult a standard, traditional textbook. You seem to be taking an approach similar to them. The ones that are in their n-th edition represent a considerable amount of experience in teaching the subject and almost certainly follow a logical order of the topics. Commented Feb 28, 2016 at 14:14
• @MichaelE2, the order followed in the textbook for the class. Commented Mar 2, 2016 at 16:36
• (1) How much time do you have? Is this a full ODE course (semester in US) or is this a small unit of DE at end of a calculus course (common in US). This affects your strategy of how much to show. Won't get to systems then. Commented Apr 8, 2018 at 15:50
• (2) If you have sufficient time, I'm puzzled why the question on order. There are some standard approaches in textbooks (with some minor variation). Why not just go in same order as assigned text? Commented Apr 8, 2018 at 15:52

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.