How do you do these questions?

I'll do the first problem to get you started.

Part (a)

We have a separable equation. Get the y term to the left side and then integrate to get

I'm using D as the integration constant rather than C since lowercase letter c was already taken.

Let's use initial condition that . This means we'll plug in t = 0 and . After doing so, solve for D

Let's plug that in and isolate y

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We end up with as our final solution. There are likely other forms to express this equation.

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Part (b)

We want y(t) to approach positive infinity.

Based on the solution in part (a), this will happen when the denominator approaches 0 from the left.

So as in which we can effectively "solve" for t showing that

If we define , then approaching T from the left side will have y(t) approach positive infinity.

This uppercase T value is doomsday. This the time value lowercase t approaches from the left when the population y(t) explodes to positive infinity.

Effectively t = T is the vertical asymptote.

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Part (c)

We're told that the initial condition is y(0) = 5 since at time 0, we have 5 rabbits. This means

Another fact we know is that y(3) = 35 because after three months, there are 35 rabbits.

Lastly, we know that c = 0.01 since the exponent of dy/dt = ky^(1.01) is 1.01; so we solve 1+c = 1.01 to get c = 0.01

We'll use y(3) = 35, c = 0.01 and to solve for k

Doing so leads to the following:

We can now compute the doomsday time value

The answer is approximately 155.67 months

-22

times both sides by 2, times () by 8, add alile terms, move terms, add alike and caculate divide both sides by 3

idk

step-by-step explanation: