9) given dx/dt = ax(x – k) & dy/dt = - bxy, solve for:
a) x in terms of t. ans. b) y in terms of t. ans.

Step-by-step explanation:

and

Step-by-step explanation:

One approach to solve this problem is first to solve   for x in terms of t and then substitute this answere in .

Starting with

 , we can use the separation of variables method.

, integrating

For solving the integral we use partial fraction decomposition method:

this leads us to the equation:

, as it can be seen, we can use this to construc a system of equations

from 1. we get and, from 2. we get .

Now

so, we get:

Now we end up with some easily solving integrals

, m is the integration cosntant.

, taking in count only positive values for x and K

then after solving for x we end up with the

.

Now for , we subtitute our last result and get

, by using the separation of variables method:

, integrating

,

using the substitution method and letting

, our integral takes the form

, using partial fraction decomposition method we end up with

, leading us to

From this: and, , thus

, integrating

, remembering

, we get

, for y positive the answer is:

a .4y4

hope this !