Let's solve the initial value problem using **Euler's Method** step by step. ### Given: \[ \frac{dy}{dx} - 5y = x \] with initial condition \( y(0) = 1 \), on interval \( [0, 1] \) with **5 steps**. First, let's rewrite the equation as: \[ \frac{dy}{dx} = x + 5y \] Let \( f(x, y) = x + 5y \). --- ## **Step 1: Set up parameters** - Interval: \( [0, 1] \) - Number of steps: \( n = 5 \) - Step size: \( h = \frac{1-0}{5} = 0.2 \) - Initial values: \( x_0 = 0 \), \( y_0 = 1 \) --- ## **Step 2: Euler's Method Formula** \[ y_{n+1} = y_n + h f(x_n, y_n) \] --- ## **Step 3: Apply Euler's Method** We will compute \( (x_i, y_i) \) for \( i = 0, 1, 2, 3, 4, 5 \): --- ### **Step 0:** - \( x_0 = 0 \) - \( y_0 = 1 \) --- ### **Step 1:** - \( x_1 = x_0 + h = 0 + 0.2 = 0.2 \) - \( f(x_0, y_0) = 0 + 5 \times 1 = 5 \) - \( y_1 = y_0 + h f(x_0, y_0) = 1 + 0.2 \times 5 = 1 + 1 = 2 \) --- ### **Step 2:** - \( x_2 = x_1 + h = 0.4 \) - \( f(x_1, y_1) = 0.2 + 5 \times 2 = 0.2 + 10 = 10.2 \) - \( y_2 = y_1 + h f(x_1, y_1) = 2 + 0.2 \times 10.2 = 2 + 2.04 = 4.04 \) --- ### **Step 3:** - \( x_3 = x_2 + h = 0.6 \) - \( f(x_2, y_2) = 0.4 + 5 \times 4.04 = 0.4 + 20.2 = 20.6 \) - \( y_3 = y_2 + h f(x_2, y_2) = 4.04 + 0.2 \times 20.6 = 4.04 + 4.12 = 8.16 \) --- ### **Step 4:** - \( x_4 = x_3 + h = 0.8 \) - \( f(x_3, y_3) = 0.6 + 5 \times 8.16 = 0.6 + 40.8 = 41.4 \) - \( y_4 = y_3 + h f(x_3, y_3) = 8.16 + 0.2 \times 41.4 = 8.16 + 8.28 = 16.44 \) --- ### **Step 5:** - \( x_5 = x_4 + h = 1.0 \) - \( f(x_4, y_4) = 0.8 + 5 \times 16.44 = 0.8 + 82.2 = 83.0 \) - \( y_5 = y_4 + h f(x_4, y_4) = 16.44 + 0.2 \times 83.0 = 16.44 + 16.6 = 33.04 \) --- ## **Summary Table** | Step \( n \) | \( x_n \) | \( y_n \) | |:---:|:---:|:---:| | 0 | 0.0 | 1.00 | | 1 | 0.2 | 2.00 | | 2 | 0.4 | 4.04 | | 3 | 0.6 | 8.16 | | 4 | 0.8 | 16.44 | | 5 | 1.0 | 33.04 | --- ## **Final Answer** Using Euler's Method with 5 steps, the estimated value of \( y(1) \) is: \[ \boxed{y(1) \approx 33.04} \] And the estimated points are: \[ (0, 1),\ (0.2, 2),\ (0.4, 4.04),\ (0.6, 8.16),\ (0.8, 16.44),\ (1.0, 33.04) \]

Euler's Method approximates the solution of differential equations by iteratively advancing from an initial point using the tangent line's slope. It provides a simple numerical approach to estimate solutions over a specified interval.