The content in your image refers to Euler's method, a numerical technique used to approximate solutions to ordinary differential equations (ODEs). Let me transcribe and solve the problem:

Problem Statement (from the image):

Use Euler's Method to find the value of $y$ at $x = 1$, given:

• The differential equation: $\frac{dy}{dx} = f(x, y)$
• Initial condition: $y(0) = 1$
• Step size: $h = 0.2$

Euler's Method Formula:

$y_{n+1} = y_n + h \cdot f(x_n, y_n)$ Where:

• $y_n$ is the current value of $y$,
• $h$ is the step size,
• $f(x_n, y_n)$ is the derivative value at $(x_n, y_n)$,
• $x_n$ is the current value of $x$.

Step-by-Step Solution:

1. Step 0 (Initial Values): $x_0 = 0, \, y_0 = 1$

2. Step 1: Compute $y_1$: $y_1 = y_0 + h \cdot f(x_0, y_0)$

   Substituting $x_0 = 0, y_0 = 1$, and $h = 0.2$: $y_1 = 1 + 0.2 \cdot f(0, 1)$

3. Step 2: Update $x_1 = x_0 + h$, and repeat the process until $x = 1$.

If you can provide the function $f(x, y)$ (e.g., $\frac{dy}{dx} = f(x, y)$), I can complete the calculations for you! Let me know if you have any details or need further clarification.

Next Steps:

1. What is the specific function $f(x, y)$?
2. How many steps need to be explicitly detailed?
3. Do you want this solved with a table showing intermediate values of $x$ and $y$?
4. Would you like a graph of the approximate solution?
5. Any additional conditions or adjustments to the initial problem?

Tip:

When using Euler's Method, smaller step sizes $h$ provide greater accuracy, but they also require more computation steps.