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We are tasked with solving a population model for a mouse population, which is governed by the differential equation:

\[
N' = 0.5N \left(1 - \frac{N}{1000}\right)
\]

The initial condition is \( N(0) = 400 \), and we are asked to use Euler's method with a step size of \( h = 0.1 \) to find the approximate population at \( t = 0.3 \).

### Euler's Method:
Euler's method is a numerical technique for solving differential equations. Given the differential equation \( N'(t) = f(t, N) \), the method updates the population \( N \) using:

\[
N_{n+1} = N_n + h \cdot f(t_n, N_n)
\]

In this case, the function \( f(t, N) = 0.5N \left(1 - \frac{N}{1000}\right) \). We'll compute the population for each time step: \( t = 0.0 \), \( t = 0.1 \), \( t = 0.2 \), and \( t = 0.3 \).

Let me calculate the population step-by-step using this method.Using Euler's method with a step size of \( h = 0.1 \), the approximate population values at different time points are as follows:

- At \( t = 0.0 \), the population \( N(0.0) = 400 \) (initial value).
- At \( t = 0.1 \), the population \( N(0.1) \approx 412.0 \).
- At \( t = 0.2 \), the population \( N(0.2) \approx 424.11 \).
- At \( t = 0.3 \), the population \( N(0.3) \approx 436.32 \).

Thus, the approximate population at \( t = 0.3 \) is **436.32**.

Would you like more details or further clarification?

Here are 5 related questions for further exploration:
1. How does the accuracy of Euler's method change with smaller step sizes?
2. What happens to the population in the long run (as \( t \to \infty \)) based on the differential equation?
3. How does the population model change if we modify the carrying capacity from 1000 to another value?
4. Can we apply the Runge-Kutta method for a more accurate solution for the same problem?
5. How would the solution change if the initial population was 600 instead of 400?

**Tip**: Euler’s method is simple but may be inaccurate over larger intervals. Reducing the step size can improve accuracy.

The rate of change of a mouse population is given by the differential equation N' = 0.5N(1 - N/1000). The population at t=0 is 400. Using Euler's method with a step size of 0.1, find the (approximate) population at t = 0.3.

Learn how to approximate the mouse population growth using Euler's method with a step size of 0.1. The differential equation N' = 0.5N(1 - N/1000) models the population dynamics, starting with an initial population of 400 at t = 0. This problem is suitable for advanced high school or undergraduate students studying numerical methods and differential equations.

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