This is (I think) too good to be hidden as comment #3 under “An Uncomfortable Teacher”.
Here’s something I handed my calculus teacher in high school:
Solve: A d2y/dx2 + B dy/dx + C y = 0
Multiplying through by dx2:
A d2y + B dy dx + C y dx2 = 0
This is a quadratic in dx:
dx = (-B dy +/- sqrt(B2 dy2 – 4AC y d2y)) / 2C y
dx/dy = (-B +/- sqrt(B2 – 4AC y d2y/dy2)) / 2C y
But d2y/dy2 = d/dy (dy/dy) = 0
dx/dy = (-B +/- sqrt(B2)) / 2C y = (-B +/- B) / 2C y
dx/dy = 0 OR (-B/C) (1/y)
EITHER x is constant (highly doubtful)
OR x = (-B/C) log y + K
IN WHICH CASE y = exp((-C/B) (x – K)) = K’ exp((-C/B) x)
Needless to say, I did not receive a convincing explanation of why this derivation was invalid.