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This afternoon was a nice reminder of why I love teaching differential equations (and teaching in general). I was explaining a new method of solving higher-order differential equations with matrix algebra, using as an example a simple second-order differential equation similar to ones the students had solved by easier methods a few chapters earlier.
After a lot of math I finally got to a quadratic that would yield two critical numbers (the eigenvalues) that would become part of the solution, and asked the students "Where have you seen this quadratic before?" then pointed to the original problem and asked "How would you have solved that problem back in Chapter 3?"
"OH!" they said, "It's the characteristic quadratic!"
In other words, an important intermediary step in this more complicated but more generalized method ends up leading to the exact same equation (but for different reasons) as the simpler but more limited method they learned a few weeks ago. And the topic of differential equations is FULL of potential "aha!" moments like this!
And then after the workshop was over one of the students said "Thanks, Barnabas; you make math fun!" which was nice.
After a lot of math I finally got to a quadratic that would yield two critical numbers (the eigenvalues) that would become part of the solution, and asked the students "Where have you seen this quadratic before?" then pointed to the original problem and asked "How would you have solved that problem back in Chapter 3?"
"OH!" they said, "It's the characteristic quadratic!"
In other words, an important intermediary step in this more complicated but more generalized method ends up leading to the exact same equation (but for different reasons) as the simpler but more limited method they learned a few weeks ago. And the topic of differential equations is FULL of potential "aha!" moments like this!
And then after the workshop was over one of the students said "Thanks, Barnabas; you make math fun!" which was nice.
