Differential what now?

For those of you here for the stories about home remodeling and pregnancy, let me explain that the field of engineering pretty much boils down to the solving of differential equations. You might therefore expect that, as someone who holds a Ph.D. in engineering, I would be capable of solving differential equations. You would be mistaken.

For a Ph.D. engineer, my math is actually pretty weak, despite the fancy equations I posted in this entry. (Some might argue, in fact, that this series of entries is actually evidence of my poor math skills, but I’m not going down that road again.) My professors can be held partly responsible for my poor math skills, because in college, my engineering professors always provided us with the solutions to the relevant differential equations instead of making us solve them ourselves. They were relying on the math department to teach us to solve the equations. Unfortunately, my math professors were pretty awful. For example, I took a class in linear algebra, taught by a Russian guy with a heavy accent. He typically started the class by writing an equation on the board and asking, “What do you think x is?”

There would be a pause, and someone would say, “Um, 1?”

“Let’s try that,” he say, and then he’d substitute 1 in for x and see if it worked out. If he ended up with something like “1=4,” he’d say, “No, 1 didn’t work. Anyone else?” Someone would suggest he try x=0, and the process would repeat. I assume that he eventually transitioned into the actual technique for solving the equation, but he never really told us when that transition occurred. So, as far as I could tell, he was teaching us to solve linear algebra equations by randomly guessing what x equals until finding something that worked. That is a pretty poor technique, as only good luck would ensure that the solution is found without having to test infinity numbers. Which would take a while.

In grad school, my engineering professors once again provided us with the solutions to all the relevant differential equations, so once again I didn’t have to know how to solve them. There was one grad school class where we were taught how to solve partial differential equations, but not really. The professor taught us the “separation of variables” technique, which is all fine and dandy if you can separate the variables. Sadly, for most real-life engineering problems, you can’t. So for most real-life engineering problems, you have to solve the differential equation numerically. (Ironically, solving an equation numerically is almost like solving an equation by randomly guessing what x equals, but using a computer so there is a chance that you will arrive at the solution in your lifetime.) Our professor taught us one numerical technique called “the finite difference method,” but he didn’t do a great job teaching us. What he did was give us a computer program which used the finite difference method to solve an equation. Then, as an assignment, he told us to use the program to solve the equation for different initial conditions. Someone asked him how many sets of initial conditions we needed to try. “At least one,” he said.

Ah.

The Doktah and I used to joke about what would happen if someone asked us how to solve a partial differential equation. “Did you try separating the variables? Oh, you can’t separate them? I can’t help you.”

But no matter how lax my professors were in teaching me how to solve differential equations, I can’t lay all the blame on them. I was in college and graduate school for nine years, so it’s possible that I should have taken some of the responsibility for my own education. Perhaps I should have taken the initiative to, oh, read about the finite difference method and actually learn it instead of just changing the initial conditions of our professor’s problem and calling it done. Because in the end, I hurt only myself.

Or did I? Because ask me how often someone asks me to solve differential equations these days. I’ll tell you. Not a lot. The skills I learned during my interminable years in college and graduate school that I actually use in my daily life are more of the critical data analysis and presentation variety and less of the higher math sort. In fact, the types of experiments I do now at work barely even resemble anything I learned in college or grad school. It most definitely is not engineering, and never requires that I solve differential equations. However, my engineering background serves me well in my ability to get to the crux of the problem at hand and in my willingness to try creative approaches in solving it.

That’s got to be worth something, right?