To solve this initial value problem using the Improved Euler's Method (also known as Heun's method), we follow a step-by-step approach. Let's assume the equation is $y' = \frac{3(e^{4t}-1)}{e^{4t}+3}$ with the initial condition $y(0) = y_0$ (we'll proceed assuming an initial value $y(0)$, which should be given explicitly, or if not, we can assign it later).

Here’s how to solve the problem:

Step 1: Define the Problem and Equation

The problem is to approximate the solution to the differential equation:

$y' = f(t, y) = \frac{3(e^{4t}-1)}{e^{4t}+3}$

over the time interval $t \in [0, 5]$ using $n = 8$ steps.

The time step size $h$ can be calculated as:

$h = \frac{t_{\text{final}} - t_{\text{initial}}}{n} = \frac{5 - 0}{8} = 0.625$

Step 2: The Improved Euler’s Method (Heun’s Method)

The formula for the Improved Euler method for each step is given by:

$y_{n+1} = y_n + \frac{h}{2} \left[ f(t_n, y_n) + f\left(t_n + h, y_n + h f(t_n, y_n)\right) \right]$

This method takes the average of the slope at the current point $(t_n, y_n)$ and an estimate at $(t_n + h, y_n + h f(t_n, y_n))$.

Step 3: Iteration Process

We will iterate from $t_0 = 0$ to $t_8 = 5$ in 8 steps with a step size of $h = 0.625$.

Let’s go step-by-step to compute the values of $y(t)$ at each point. First, I will calculate the values using the formula above for each step.

Step 3.1: Initial Condition

At $t_0 = 0$, we are given $y(0) = y_0$. Let’s proceed with that.

Step 3.2: Apply Heun's Method

For each step, we calculate the next value of $y_{n+1}$ using the Improved Euler's formula. I'll carry out the calculations and show the progression for each step below:

1. Step 0 (Initial point): $t_0 = 0$, $y_0 = y(0)$ (assumed or given initial condition).
2. Step 1: $t_1 = 0.625$
3. Step 2: $t_2 = 1.25$
4. Step 3: $t_3 = 1.875$
5. Step 4: $t_4 = 2.5$
6. Step 5: $t_5 = 3.125$
7. Step 6: $t_6 = 3.75$
8. Step 7: $t_7 = 4.375$
9. Step 8: $t_8 = 5.0$

I will now compute the exact $y(t_n)$ values numerically using Python for better precision.

Let me do the calculations:The solution using Improved Euler's method with $n = 8$ steps gives the following approximate values of $y(t)$ at the specified time points:

• $y(0.00) = 0.000$
• $y(0.625) = 0.691$
• $y(1.25) = 2.294$
• $y(1.875) = 4.142$
• $y(2.5) = 6.015$
• $y(3.125) = 7.889$
• $y(3.75) = 9.764$
• $y(4.375) = 11.639$
• $y(5.00) = 13.514$

Thus, at $t = 5$, the approximate value of $y(5)$ is 13.514 using this method.

Would you like more details on any of these calculations or steps?

Here are some follow-up questions that could expand your understanding:

1. How does the Improved Euler's method compare with the standard Euler's method?
2. What is the effect of increasing or decreasing the step size $h$?
3. How does the behavior of $y(t)$ change as $t$ increases?
4. Can we apply this method to more complex or nonlinear systems?
5. How accurate is the solution if we use more steps?

Tip: The Improved Euler's method is much more accurate than the basic Euler method because it averages the slope at two points, reducing error in each step.