To solve this problem using Euler's method, let's first understand the system of differential equations provided and the initial condition. Euler's method allows us to approximate solutions for differential equations step by step.

The system of equations:

$\frac{dx}{dt} = x + y$ $\frac{dy}{dt} = 6y$

Initial conditions:

$x(0) = 3, \quad y(0) = -4$ The step size is given as $\Delta t = \frac{1}{3}$.

Euler's Method Formula:

For each step, Euler's method updates $x$ and $y$ as follows: $x_{n+1} = x_n + \Delta t \cdot f(x_n, y_n)$ $y_{n+1} = y_n + \Delta t \cdot g(y_n)$ where $f(x, y) = x + y$ and $g(y) = 6y$ are the right-hand sides of the equations.

Step 1: First Iteration ($x_1$, $y_1$)

At $t_0 = 0$, we start with: $x_0 = 3, \quad y_0 = -4$

1. Compute $f(x_0, y_0) = x_0 + y_0 = 3 + (-4) = -1$.

   • Then update $x_1$: $x_1 = x_0 + \Delta t \cdot f(x_0, y_0) = 3 + \frac{1}{3} \cdot (-1) = 3 - \frac{1}{3} = 2.67$

2. Compute $g(y_0) = 6y_0 = 6 \cdot (-4) = -24$.

   • Then update $y_1$: $y_1 = y_0 + \Delta t \cdot g(y_0) = -4 + \frac{1}{3} \cdot (-24) = -4 - 8 = -12$

So, after the first iteration: $x_1 = 2.67, \quad y_1 = -12$

Step 2: Second Iteration ($x_2$, $y_2$)

Now, using $x_1 = 2.67$ and $y_1 = -12$:

1. Compute $f(x_1, y_1) = x_1 + y_1 = 2.67 + (-12) = -9.33$.

   • Then update $x_2$: $x_2 = x_1 + \Delta t \cdot f(x_1, y_1) = 2.67 + \frac{1}{3} \cdot (-9.33) = 2.67 - 3.11 = -0.44$

2. Compute $g(y_1) = 6y_1 = 6 \cdot (-12) = -72$.

   • Then update $y_2$: $y_2 = y_1 + \Delta t \cdot g(y_1) = -12 + \frac{1}{3} \cdot (-72) = -12 - 24 = -36$

So, after the second iteration: $x_2 = -0.44, \quad y_2 = -36$

Final Answer:

• $x_1 = 2.67$
• $y_1 = -12$
• $x_2 = -0.44$
• $y_2 = -36$

Would you like more details or have any questions?

Here are 5 related questions:

1. How does the accuracy of Euler's method depend on the step size $\Delta t$?
2. What are the limitations of using Euler’s method for solving differential equations?
3. How can we improve the accuracy of Euler’s method without changing the step size?
4. What other numerical methods can be used for solving systems of differential equations?
5. Can Euler’s method be applied to nonlinear systems of differential equations?

Tip: The smaller the step size $\Delta t$, the more accurate Euler’s method is, but at the cost of more computations.