The key model for growth (or decay when c < 0) is dy/dt = c y(t)
The next model allows a steady source (constant s in dy/dt = cy + s )
The solutions include an exponential e^ct (because its derivative brings down c)

So growth forever if c is positive and decay if c is negative
A neat model for the population P(t) adds in minus sP^2 (so P won't grow forever)

This is nonlinear but luckily the equation for y = 1/P is linear and we solve it

Population P follows an "S-curve" reaching a number like 10 or 11 billion (???)
Great lecture but Professor Strang should have written e^-ct in the last formula