I have a multipart problem, but I'm unclear on how to approach the second half. I integrated the equation dL/dt=a(Le - L ) (where Le = eventual length)...then next part of the problem says to evaluate the new constant of integration (the integrated algebraic result for L(t)) by applying the initial condition (L=0 when t=0).
dL/dt = a Le - a L
dL = a (Le-L) dt
let x = a Le - a L
dx = -a dL
so
dx/a = -x dt
dx/x = --a dt
ln x = -a t + c
ln (a Le - a L) = - a t + c
a Le - a L = e^(-at+c) = e^c e^-at = ke^-at
so
when t = 0
a Le - 0 = k e^0 = k
so
a Le = k
so
a Le - a L = a Le e^-at
1 - L/Le = e^-at
L/Le = 1 - e^at