To solve the differential equation using Euler's method, we'll proceed step by step. The differential equation is given as:

$\frac{dy}{dx} = 2 - 1.1yx$

with the initial condition $y(0) = 1$ over the interval $x = 0$ to $x = 2$ with step size $h = 0.5$.

Step-by-Step Procedure:

1. Initial Values: $x_0 = 0, \quad y_0 = 1$

2. Euler's Formula: $y_{n+1} = y_n + h \cdot f(x_n, y_n)$ where $f(x, y) = 2 - 1.1yx$.

3. Calculate the next values iteratively:

   • Step 1: $x_1 = 0.5$ $f(x_0, y_0) = 2 - 1.1 \cdot y_0 \cdot x_0 = 2 - 1.1 \cdot 1 \cdot 0 = 2$ $y_1 = y_0 + h \cdot f(x_0, y_0) = 1 + 0.5 \cdot 2 = 1 + 1 = 2$

   • Step 2: $x_2 = 1.0$ $f(x_1, y_1) = 2 - 1.1 \cdot y_1 \cdot x_1 = 2 - 1.1 \cdot 2 \cdot 0.5 = 2 - 1.1 = 0.9$ $y_2 = y_1 + h \cdot f(x_1, y_1) = 2 + 0.5 \cdot 0.9 = 2 + 0.45 = 2.45$

   • Step 3: $x_3 = 1.5$ $f(x_2, y_2) = 2 - 1.1 \cdot y_2 \cdot x_2 = 2 - 1.1 \cdot 2.45 \cdot 1 = 2 - 2.695 = -0.695$ $y_3 = y_2 + h \cdot f(x_2, y_2) = 2.45 + 0.5 \cdot (-0.695) = 2.45 - 0.3475 = 2.1025$

   • Step 4: $x_4 = 2.0$ $f(x_3, y_3) = 2 - 1.1 \cdot y_3 \cdot x_3 = 2 - 1.1 \cdot 2.1025 \cdot 1.5 = 2 - 3.467625 = -1.467625$ $y_4 = y_3 + h \cdot f(x_3, y_3) = 2.1025 + 0.5 \cdot (-1.467625) = 2.1025 - 0.7338125 = 1.3686875$

Final Solution:

The approximate solution for the initial value problem over the interval $x = 0$ to $x = 2$ using Euler's method with $h = 0.5$ is:

$y(2) \approx 1.3687$

Would you like more details on any part of the process, or do you have any other questions?

Additional Questions:

1. How does the accuracy of Euler's method compare to higher-order methods like the Runge-Kutta method?
2. How would the solution change if the step size $h$ were smaller, say $h = 0.1$?
3. Can Euler's method be applied to systems of differential equations, and if so, how?
4. How does the stability of Euler's method affect its use for stiff differential equations?
5. How can you estimate the error introduced by using Euler's method?

Tip:

To improve the accuracy of the numerical solution, consider using smaller step sizes or higher-order methods like the Runge-Kutta method.