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?

  • 5
    $\begingroup$ 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. $\endgroup$
    – user1815
    Commented Feb 28, 2016 at 14:14
  • 1
    $\begingroup$ @MichaelE2, the order followed in the textbook for the class. $\endgroup$
    – vonbrand
    Commented Mar 2, 2016 at 16:36
  • $\begingroup$ (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. $\endgroup$
    – guest
    Commented Apr 8, 2018 at 15:50
  • $\begingroup$ (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? $\endgroup$
    – guest
    Commented Apr 8, 2018 at 15:52

2 Answers 2


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.


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.


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