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The given problem is a logistic differential equation:

\[
\frac{dP}{dt} = 0.1 P \left( 1 - \frac{P}{2000} \right)
\]

with the initial condition \( P(0) = 100 \).

### a) Solving the Differential Equation

This is a standard logistic growth model. The general solution of a logistic equation of the form:

\[
\frac{dP}{dt} = r P \left(1 - \frac{P}{K}\right)
\]

where \(r = 0.1\) and \(K = 2000\), is:

\[
P(t) = \frac{K}{1 + \left( \frac{K - P_0}{P_0} \right) e^{-rt}}
\]

Substituting the known values:

- \(K = 2000\) (carrying capacity),
- \(P_0 = 100\) (initial population),
- \(r = 0.1\) (growth rate),

we get the specific solution:

\[
P(t) = \frac{2000}{1 + \left( \frac{2000 - 100}{100} \right) e^{-0.1t}} = \frac{2000}{1 + 19 e^{-0.1t}}
\]

### Finding Population at \(t = 20\)

To find the population when \(t = 20\), we substitute \(t = 20\) into the solution:

\[
P(20) = \frac{2000}{1 + 19 e^{-0.1 \times 20}} = \frac{2000}{1 + 19 e^{-2}} 
\]

Using \(e^{-2} \approx 0.1353\):

\[
P(20) = \frac{2000}{1 + 19(0.1353)} = \frac{2000}{1 + 2.5707} = \frac{2000}{3.5707} \approx 560.29
\]

So, the population after 20 units of time is approximately 560.

---

### b) Finding When the Population Reaches 1200

To find when the population reaches 1200, we solve the equation:

\[
1200 = \frac{2000}{1 + 19 e^{-0.1t}}
\]

First, solve for \(e^{-0.1t}\):

\[
1 + 19 e^{-0.1t} = \frac{2000}{1200} = 1.6667
\]

\[
19 e^{-0.1t} = 1.6667 - 1 = 0.6667
\]

\[
e^{-0.1t} = \frac{0.6667}{19} \approx 0.03509
\]

Now, take the natural logarithm of both sides:

\[
-0.1t = \ln(0.03509) \approx -3.350
\]

\[
t = \frac{-3.350}{-0.1} \approx 33.5
\]

So, the population reaches 1200 at approximately \(t = 33.5\) units of time.

---

### Summary of Solutions:
- The population at \(t = 20\) is approximately 560.
- The population reaches 1200 at \(t \approx 33.5\).

Would you like more details on any of the steps? Here are some related questions to explore:

1. How would the solution change if the growth rate \(r\) were different?
2. Can we find the limiting behavior of \(P(t)\) as \(t \to \infty\)?
3. How does the carrying capacity \(K\) affect the solution?
4. What is the significance of the exponential decay term \(e^{-rt}\)?
5. Can we derive the general logistic equation from basic principles of population dynamics?

Tip: For logistic models, the population always approaches the carrying capacity \(K\) over time, even if the initial population is far from it.

Escriba la solución del problema con valor inicial: dP/dt = 0.1P(1 - P/2000), P(0) = 100. Utilice la solución para hallar la población cuando t = 20. ¿Cuándo la población alcanza 1200?

Learn how to solve a logistic growth equation dP/dt = 0.1P(1 - P/2000) with an initial population P(0) = 100. Find the population at t = 20 and determine when the population will reach 1200. Detailed steps include using the logistic equation and solving for exponential decay.

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