Question

I am not sure how to set this equation up, if you could please help me I would greatly appreciate it. Thank you. The rate of change of the number of wolves W(t) in a population is proportional to the quantity 1500-W(t), where t> 0 is time measured in years and k is the constant of proportionality. Assume as an initial condition that when t=0 the wolf population is 500

Answer 1

\[W'(t) = k(1500-W(t))\] The rate of change, W'(t) is proportional to 1500-W(t), so that gives us the right hand side. Solve the equation, then plug in the initial condition that W(0) = 500 and find the value of k which makes it so.

Answer 2

would k=1/1000

Answer 3

I think you'll need to integrate this... first make it easier to do so\[\frac{ dW }{dt } = k(1500-W)\]
\[\frac{ dW }{(1500-W) } = (k )dt\]
\[\frac{ 1 }{(1500-W) }dW = (k )dt\]
now integrate both sides.

Answer 4

How do you integrate both sides

Answer 5

Integrate the left w.r.t. W, right w.r.t. to t.

Try a substitution on the left, let u = 1500-W.

Answer 6

\[\int\limits \frac{ 1 }{(1500-W) }dW = \int\limits (k )dt\] substitute u = 1500-W, so du = -1 dW \[-\int\limits\limits \frac{ 1 }{u }du =k \int\limits\limits dt\]