Question
Consider the differential equation:
\[ \frac{dy}{dx} = xy – 1 \]
Given that \( y = 2 \) when \( x = 1 \), use Euler’s method with step size \( h = 0.1 \) to find the approximate value of \( y \) when \( x = 1.5 \).
▶️ Answer/Explanation
Detailed Solution
Step 1: Euler’s Method Formula
The iterative formula for Euler’s method is:
\[ y_{n+1} = y_n + h \cdot f(x_n, y_n) \]
where \( f(x, y) = xy – 1 \) and \( h = 0.1 \).
Step 2: Compute Successive Approximations
| Step | \( x_n \) | \( y_n \) | \( f(x_n, y_n) \) | \( y_{n+1} \) |
|---|---|---|---|---|
| 1 | 1.0 | 2.0000 | (\(1 \times 2 – 1 = 1\)) | \(2.0000 + 0.1(1) = 2.1000\) |
| 2 | 1.1 | 2.1000 | (\(1.1 \times 2.1 – 1 = 1.31\)) | \(2.1000 + 0.1(1.31) = 2.2310\) |
| 3 | 1.2 | 2.2310 | (\(1.2 \times 2.231 – 1 = 1.6772\)) | \(2.2310 + 0.1(1.6772) = 2.3987\) |
| 4 | 1.3 | 2.3987 | (\(1.3 \times 2.3987 – 1 = 2.1183\)) | \(2.3987 + 0.1(2.1183) = 2.6105\) |
| 5 | 1.4 | 2.6105 | (\(1.4 \times 2.6105 – 1 = 2.6547\)) | \(2.6105 + 0.1(2.6547) = 2.8760\) |
Final Answer: When \( x = 1.5 \), \( y \approx 2.88 \) (rounded from 2.87603).
……………………………….Markscheme……………………………….
- Application of Euler’s method: Correct iterative formula used.
- Step-by-step calculations: Each step follows Euler’s formula.
- Final approximation: \( y \approx 2.88 \).