Question

If \( \dfrac{dy}{dx} = xy + y^2 \), \( y(0) = 1 \), and \( h = 0.1 \), then the value of \( y(0.1) \) using Euler’s method is:

• A. 0.1
• B. 0.11
• C. 1.01
• D. 1.1

Hint:

Euler's method approximates the next value using the slope at the current point: \( y_{n+1} = y_n + h \cdot f(x_n, y_n) \).

Answer 1

The Correct Option is D

To solve the given differential equation using Euler's method, follow these steps:

Step 1: Set Initial Conditions

Given: \( y(0) = 1 \), \( h = 0.1 \).

Step 2: Apply Euler's Method Formula

The formula for Euler's method is: \( y_{n+1} = y_n + h \cdot f(x_n, y_n) \).

Where \( f(x, y) = xy + y^2 \). For the first step:

\( x_0 = 0 \), \( y_0 = 1 \).

Step 3: Calculate \( y_1 \)

Calculate the first derivative at \((x_0, y_0)\):

\( f(0, 1) = 0 \cdot 1 + 1^2 = 1 \).

Apply the formula:

\( y_1 = y_0 + h \cdot f(x_0, y_0) = 1 + 0.1 \cdot 1 = 1.1 \).

Thus, after applying Euler's method for one step, \( y(0.1) \approx 1.1 \).

Conclusion:

The value of \( y(0.1) \) using Euler's method is approximately 1.1.