College Math Teaching

September 25, 2010

The Peril of Teaching Advanced Techniques to Intellectually Immature Students

Filed under: derivatives, differential equations, mathematics education, student learning — collegemathteaching @ 9:41 pm

Ok, I told my differential equations class to solve:
y' = y(y-2) with initial condition y(0) = 1
This is straight forward right? I expected the “good” student to separate the variables:
dy/(y(y-2)) = dt and integrate; the left hand side can be done by an easy integration; one can use partial fractions but I allow them to use calculators that have symbolic calculus operations on it.

Now of course, you had the bad student mistake: integrating the right hand side with respect to y :
(1/3)y^3 -y^2 + C = y but you will always have some clueless students.

But what surprised me was this: some students used the Bernoulli change of variable method:
y' + 2y =y^2 which leads to y^{-2}y' + 2y^{-1}=1 and then setting z = y^{-1} which leads to: z' = -y^{-2}y'
and that gives z' + 2z = 1 which has solution z = 1/2 + ke^{-2t} and then y = 2/(1 + ke^{-2t})
Those who did this got it mostly right, though most made algebraic errors along the way.

So I am glad that they knew the more advanced technique but why use it here? 🙂

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