Ok, I've been doing BC review tests to prepare for the AP exam. I came across this free response problem and I have no idea how to do it, can someone please help me with this?

It gives the differential equations:
dP/dt = .03P(30-P)

and it asks "Find the general solution to the differential equation".

The answer is P(t)= 30Ce^0.9t/(1+Ce^c.9t)

Any help is greatly appreciated, thank you!

Of course! I'll be glad to help you with the problem. Solving differential equations can be a bit challenging, but we can break it down step by step to find the general solution.

First, let's rewrite the given differential equation:

dP/dt = 0.03P(30 - P)

We can start by separating the variables. To do this, we'll divide both sides by P(30 - P) to get:

1/(P(30 - P)) * dP/dt = 0.03

Now, we integrate both sides with respect to t. The left side requires a bit of manipulation using the partial fraction decomposition method. We can decompose the expression as follows:

1/(P(30 - P)) = A/P + B/(30 - P)

Multiplying through by P(30 - P), we get:

1 = A(30 - P) + BP

Expanding and collecting like terms:

1 = (30A - A*P) + BP

Comparing coefficients, we have:

-AP + BP = 0 (coefficient of P term)
A(30) = 1 (constant term)

From the first equation, we can rearrange it to solve for A/B:

A/B = P/B

And from the second equation, we get:

A(30) = 1
30A = 1
A = 1/30

Substituting this value back into our expression for A/B:

(1/30)/B = P/B

Canceling out the B terms, we have:

1/30 = P

Now we can rewrite the left side of the differential equation as:

(1/30)*dP/dt

Integrating both sides with respect to t, we get:

∫(1/30)dP = 0.03 ∫dt

(1/30) ∫dP = 0.03t + C1

ln(P) = 0.03t + C1

Exponentiating both sides, we have:

P = e^(0.03t + C1)

Now, let's simplify the right side by using the properties of exponents. We can write it as:

P = e^(0.03t) * e^(C1)

Let's define a constant C as e^(C1):

P = e^(0.03t) * C

Now, we need to find the value of C. To do this, we'll use the initial condition given in the problem. You haven't mentioned an initial condition, so we'll proceed assuming that P(0) = 0.

When t = 0, we have:

P(0) = e^(0.03(0)) * C
P(0) = C

Since P(0) should be 30 (as given in the answer), we can now substitute in the correct value:

C = 30

So, the general solution to the differential equation is:

P(t) = 30 * e^(0.03t)

And there you have it! The general solution is similar to the answer you provided, but there is a slight mistake in the exponent. It should be 0.03t instead of 0.9t.