We are tasked with solving the given initial value problem using Euler's method. Here's the problem statement extracted from the image:

• Differential equation: $\frac{dy}{dx} = 1 - y$, with the initial condition $y(0) = 0$.
• Step size: $h = 0.1$.
• We need to compute $y(0.2)$.

Steps for Euler's Method

1. Formula: $y_{n+1} = y_n + h \cdot f(x_n, y_n)$ where $f(x, y) = 1 - y$ is the right-hand side of the differential equation.

2. Initialization:

   • Start with $x_0 = 0$ and $y_0 = 0$.

3. Iterations:

   • Compute $y_1$ using $x_1 = x_0 + h = 0.1$.
   • Compute $y_2$ using $x_2 = x_1 + h = 0.2$.

Computation

1. Step 1 (from $x_0 = 0$ to $x_1 = 0.1$): $f(x_0, y_0) = 1 - y_0 = 1 - 0 = 1$ $y_1 = y_0 + h \cdot f(x_0, y_0) = 0 + 0.1 \cdot 1 = 0.1$

2. Step 2 (from $x_1 = 0.1$ to $x_2 = 0.2$): $f(x_1, y_1) = 1 - y_1 = 1 - 0.1 = 0.9$ $y_2 = y_1 + h \cdot f(x_1, y_1) = 0.1 + 0.1 \cdot 0.9 = 0.1 + 0.09 = 0.19$

Final Answer

Using Euler's method, $y(0.2) \approx 0.19$.

Would you like to see further steps, such as error analysis or a comparison to the exact solution?

Related Questions

1. What is the error of Euler's method for this computation?
2. How would a smaller step size $h$ affect the accuracy of the solution?
3. Can we apply other numerical methods like the Runge-Kutta method for better accuracy?
4. What is the exact solution to the differential equation, and how does it compare to the approximate solution?
5. How does Euler's method behave for equations with rapidly changing solutions?

Tip:

Always check how the error accumulates in numerical methods by comparing to the exact or refined solutions when possible.