Your course sounds very much like Harvard's 21b, which I've taught a few times. I actually really like the way we do the PDE's unit, although I think two weeks is not enough time and (consequently) the students don't like it so much. Perhaps you can adapt the following method to your satisfaction. Our textbook, by the way, is Otto Bretscher's "Linear Algebra with Applications", though the stuff on PDE's is not there (it's here instead).

Some pre-explanation: a few weeks earlier in the course we cover systems of first-order linear ODE's, namely,

$$\frac{d\vec{x}}{dt} = A \vec{x},$$

with A a matrix. We explain how this can be reduced to n simple integrations by diagonalizing A and then writing the general solution as a linear combination of eigenvectors scaled by the various resulting exponentials. This should impress on the students the importance of the idea of solving an equation of the form "derivative(x) = operator(x)" by diagonalizing "operator". (This part of the course IS in Bretscher's book.)

When we get to the PDE's unit, we have already talked about Fourier series a bit; no one quite gets why, but that's because we haven't done the application yet. Then we slap down the heat equation,

$$\frac{df}{dt} = \mu \frac{d^2f}{dx^2},$$

and observe that now "operator" is $d^2/dx^2$, but the same idea applies. We solve for its eigenvectors and lo! they are the trigonometric functions in x. Then we do the simple integrations in t and the resulting "linear combination" is a Fourier series (whose coefficients are decaying exponentials), which turn out to be useful after all. We even go on to do the wave equation in the same way. Note how the variables got separated so naturally.

Like I said, I love the way the process is reduced to an idea in linear algebra having nothing to do with sophisticated notions in analysis or tricks involving calculus. Those are for courses truly devoted to PDE's; in this course, it is enough to just explain why separating variables even happens.

Your course sounds very much like Harvard's 21b, which I've taught a few times. I actually really like the way we do the PDE's unit, although I think two weeks is not enough time and (consequently) the students don't like it so much. Perhaps you can adapt the following method to your satisfaction. Our textbook, by the way, is Otto Bretscher's "Linear Algebra with Applications", though the stuff on PDE's is not there (it's here instead).

Some pre-explanation: a few weeks earlier in the course we cover systems of first-order linear ODE's, namely,

$$\frac{d\vec{x}}{dt} = A \vec{x},$$

with A a matrix. We explain how this can be reduced to n simple integrations by diagonalizing A and then writing the general solution as a linear combination of eigenvectors scaled by the various resulting exponentials. This should impress on the students the importance of the idea of solving an equation of the form "derivative(x) = operator(x)" by diagonalizing "operator". (This part of the course IS in Bretscher's book.)

When we get to the PDE's unit, we have already talked about Fourier series a bit; no one quite gets why, but that's because we haven't done the application yet. Then we slap down the heat equation,

$$\frac{df}{dt} = \mu \frac{d^2f}{dx^2},$$

and observe that now "operator" is $d^2/dx^2$, but the same idea applies. We solve for its eigenvectors and lo! they are the trigonometric functions in x. Then we do the simple integrations in t and the resulting "linear combination" is a Fourier series (whose coefficients are decaying exponentials), which turn out to be useful after all. We even go on to do the wave equation in the same way. Note how the variables got separated so naturally.

Like I said, I love the way the process is reduced to an idea in linear algebra having nothing to do with sophisticated notions in analysis or tricks involving calculus. Those are for courses truly devoted to PDE's; in this course, it is enough to just explain why separating variables even happens.

EDIT: Upon rereading, I feel like I should preempt the inevitable objection that the hyperbolic trig functions (equivalently, exponential functions) are also eigenvectors of $d^2/dx^2$. Of course, we throw those out because they don't satisfy the boundary conditions we are using, which are generally that $f(0,t) = f(1,t) = 0$ or similar. Explaining this is usually the cause of moderate distress among the students, so recently, I noticed that we just ignore it.