Here's a few things I think are worthwhile:
- deeper substitutions, solve $y''+y=0$ by explicit substitution and integration. It's a nice think to touch base with as you go on in the course.
- theory of n-th order linear systems. How to capture linear independence. Discuss the distinction between LI for solution sets vs. arbitrary sets of functions. Abel's formula holding for the n-th order problem is a bit shocking really. Spend some time on the Wronskian and get them thinking about how cool it is we can use finite dimensional linear algebra in function space and get away with it.
- hand in hand with last bullet point, operators theory. Discuss and develop the theory of smooth operators. Noncommutativity is easy to exhibit, yet we largely solve problems of the form $L[y]=f$ where $L \in \mathbb{R}[D]$ and $D=d/dt$. Why can we factor such $L$. What about when $L = L_1L_2\cdots L_k$ and $L_1,\dots , L_k$ are not mere polynomials in the differentiation operator? How to solve such a problem? It's a lost art, but it's easy.
- again hand in hand in hand with the bullets above, the annihilator method. Rather than just forcing them to memorize how to construct the particular solution, this gives them a way to derive it. Also, gives us a good reason to care about larger than order two problems since it is entirely possible for a spring mass problem to beg us solving a $4$-th order or higher ODE.
- Variation of parameters is general whereas undetermined coefficients, obviously not. Notice from the operator approach you can even frame Cauchy-Euler problems as polynomials in $td/dt$ and much of the discussion can be mirrored. Or, you could be boring and just plug in $x^r$ and $e^{rt}$ and do the less inspired calculus.
- So much to see in systems! Notice the fact that $y''+2y'+y=0$ has solution $y = c_1e^{_t}+c_2te^{-t}$ already indicates the corresponding system obtained by reduction of order is not diagonalizable. Furthermore, much riches can be found simply by dwelling on the question of that mysterious $t$ in the solution here.
- matrix exponential. Need I say more?
- PDEs are quite challenging, but, for better students perhaps this is an opportunity to remind them where much of our soul searching stems from in analysis. Fourier series beg questions about interchange of limits and illustrate why details matter. Ironically, details of analysis matter less to the problems you're likely to cover in this course. If you can work in the solution via Laplace transform material I think it's a win here. I sometimes succeed.
- complex analysis. Comment on connections when you get a chance.
- last but certainly not least, the differentiate the parameter trick. Why does it work? Where does it work? Specifically, $\frac{d}{dr}e^{tr} = te^{tr}$ and $\frac{d}{dr}(x^r) = \ln(x)x^r$...
If you're really interested and happen to have students who have had abstract algebra before your class, I have additional suggestions, but they're way outside the box.
Edit: ok, so what's outside the box? How about this, to solve any constant coefficient differential equation over an algebra we can simply select the component functions of the exponential over the characteristic extension algebra. For example, to solve $y''+y=0$ over $\mathbb{R}$ we work on $\mathcal{A} = \mathbb{R}[k]/(k^2+1)$ where $k^2=-1$. In this context,
$$ e^{kt} = 1+kt+\frac{1}{2}k^2t^2+ \cdots = \cos(t)+k \sin t $$
thus the solution is formed by the special functions $\cos t$ and $\sin t$ of the characteristic extension algebra $\mathcal{A}$. But, lest you think this is just complexification. Notice this technique works for any problem. For example, $y''-2y'+y=0$ suggests $\mathcal{A} = \mathbb{R}[k]/(k^2-2k+1)$ which is to say $(k-1)^2 = 0$. Once more consider, the map $t \mapsto e^{kt}$ and calculate:
$$ e^{kt} = e^{(k-1)t+t} =e^te^{(k-1)t} = e^t(1+(k-1)t) = (1-t)e^t+kte^t $$
thus $(1-t)e^t, te^t$ form a fundamental solution set. I have proof that the component functions of the exponential always form an $\mathcal{A}$-LI solution set even when the solution set is a module (as is the case when the base algebra is not a field, in both of my examples the base algebra was just $\mathbb{R}$). If you understand what I am saying here, you'll realize I just put forth a method which solves all n-th order ODEs via the same technique. Studying differential equations over algebras which are not fields brings new challenges if you wish to continue down this path. For example, we can show the differential equation $(D-1)(D-j)[y]=0$ where $D = d/dz$ and $j^2=1$ in the hyperbolic numbers $\mathcal{H} = \mathbb{R}\oplus j \mathbb{R}$ has solutions $e^z$ and $e^{jz}$ which are linearly dependent. It can be shown that:
$$ w = c_2e^{jz}+c_1e^{jz}\left( z+
\sum_{n=1}^{\infty} \frac{1-j}{(n-1)!(n+1)} z^{n+1}\right). $$
is the general solution. You see, the distinct characteristic values $\lambda_1 = 1$ and $\lambda_2 = j$ behave as if they are repeated... almost...The reason for this bizarre behavior is simply that $(1-j)(1+j)=0$ so the difference of the characteristic values is a zero divisor of the algebra. My experience with these things is that zero divisors bring rather different structure than that we've grown accustomed to see over the fields $\mathbb{R}$ or $\mathbb{C}$. I have more to say in my joint paper with Nathan BeDell.
The abstract algebra used here is mostly the extension of Kronecker's Theorem to polynomials which need not be irreducible. The key construction is just that $\mathcal{A}[k]/(p(k))$ for an $n$-order monic polynomial $p$ is in fact a free $\mathcal{A}$-module with basis $1,k,k^2, \dots , k^{n-1}$. In the case $\mathcal{A}=\mathbb{R}$ this is just an $n$-dimensional vector space. Beyond that, I had to develop a bit of calculus over an algebra. Details are found in my papers on these topics.
