Find y(0.1) using Euler Method, dy/dx = y - x, y(0) = 2, h = 0.02.

Steps to Solve

1. Determine the number of steps
   We want to find $y(0.1)$ using a step size of $h = 0.02$.
   To find the total number of steps, calculate:
   $$ n = \frac{0.1 - 0}{0.02} = 5 $$
   So, we will perform 5 iterations of Euler's method.

2. Initialize variables
   Set the initial values:

• $x_0 = 0$,
• $y_0 = 2$.

3. Euler's Method Iteration
   For each step, use the formula:
   $$ y_{n+1} = y_n + h \cdot f(x_n, y_n) $$
   where $f(x, y) = y - x$.
   Now compute each iteration:

• Iteration 1:

  • Calculate $f(x_0, y_0) = y_0 - x_0 = 2 - 0 = 2$.
  • Update $y_1$:
    $$ y_1 = y_0 + h \cdot f(x_0, y_0) = 2 + 0.02 \cdot 2 = 2.04 $$
  • Update $x_1$:
    $$ x_1 = x_0 + h = 0 + 0.02 = 0.02 $$

• Iteration 2:

  • Calculate $f(x_1, y_1) = y_1 - x_1 = 2.04 - 0.02 = 2.02$.
  • Update $y_2$:
    $$ y_2 = y_1 + h \cdot f(x_1, y_1) = 2.04 + 0.02 \cdot 2.02 = 2.0804 $$
  • Update $x_2$:
    $$ x_2 = x_1 + h = 0.02 + 0.02 = 0.04 $$

• Iteration 3:

  • Calculate $f(x_2, y_2) = y_2 - x_2 = 2.0804 - 0.04 = 2.0404$.
  • Update $y_3$:
    $$ y_3 = y_2 + h \cdot f(x_2, y_2) = 2.0804 + 0.02 \cdot 2.0404 = 2.120808 $$
  • Update $x_3$:
    $$ x_3 = x_2 + h = 0.04 + 0.02 = 0.06 $$

• Iteration 4:

  • Calculate $f(x_3, y_3) = y_3 - x_3 = 2.120808 - 0.06 = 2.060808$.
  • Update $y_4$:
    $$ y_4 = y_3 + h \cdot f(x_3, y_3) = 2.120808 + 0.02 \cdot 2.060808 = 2.24163216 $$
  • Update $x_4$:
    $$ x_4 = x_3 + h = 0.06 + 0.02 = 0.08 $$

• Iteration 5:

  • Calculate $f(x_4, y_4) = y_4 - x_4 = 2.24163216 - 0.08 = 2.16163216$.
  • Update $y_5$:
    $$ y_5 = y_4 + h \cdot f(x_4, y_4) = 2.24163216 + 0.02 \cdot 2.16163216 = 2.3857518272 $$
  • Update $x_5$:
    $$ x_5 = x_4 + h = 0.08 + 0.02 = 0.1 $$

4. Final value
   The approximated value at $y(0.1) = y_5$.