visualizing logistic population growth

edit

This is a followup question to this question where @mforets helped me translate some Ti Nspire code to Sagemath code. I now have following,

t = var('t')
y = function('y')(t)
ye = desolve(diff(y,t) == 2*10^(-5)*y*(1500-y), y, ics=[0,50])
ye = ye*3/100
yt = solve(ye.simplify_log(), y)
show(expand(yt))

Now I'm interested to visualize this result, reproducing some lecture notes. I looked at the examples given in the Sage Quickstart for Differential Equations, but I cold not reproduce what was there with my example.

My lecture notes has a lot like this that I'm aim at

By copying the right hand side (code below figure) manually I can get this, but I feel like I'm missing something;

P = plot((1500*e^(3/100*t)/(e^(3/100*t) + 29)),t,0,250, color='red')
L = line([(0,1500), (260,1500)], linestyle=":", color="grey")
show(P+L, figsize=5)