## MIT 1803 lecture 8

This lecture gave me insight into approaching learning and remembering formulas.  This formula

$a \cos \theta + b \sin \theta = C \cos (\theta + \phi)$

was proved three separate ways in class, but is best remembered  by a single diagram.

(Note that the <a, b> notation is for a complex vector).  To find the proof, examine $(a-ib) \cdot (\cos \theta + i \sin \theta)$, and take the real part.

This was the last lecture on first order linear ODEs, so he reviewed the three equations in order of most to least general.

$y' = ky = kq(t)$

$y' = ky = q(t)$

$y' = p(t)y = q(t)$

Noting that these equations show up in “life” (biology, economics) when $k>0$ and in “inanimate” settings when $k<0$.   Finding a “steady-state” and tranient solution only applies in the first case.