Let \( y=f(x) \) be the solution to the differential equation \( \frac{d y}{d x}=x+y \) with the initial condition \( f(1)=2 \). What is the approximation for \( f(2) \) if Euler's method is used, starting at \( x=1 \) with a step size of 0.5 ?

To use Euler's method to approximate \( f(2) \), we start at the initial condition \( (1, 2) \) and apply the differential equation given by \( \frac{dy}{dx} = x + y \). 1. At \( x=1 \), \( y=f(1)=2 \): \[ \frac{dy}{dx} = 1 + 2 = 3 \] The next value of \( y \) after a step of 0.5 is: \[ y_1 = y_0 + f'(x_0) \cdot \Delta x = 2 + 3 \cdot 0.5 = 2 + 1.5 = 3.5 \] Thus, at \( x=1.5 \), \( f(1.5) \approx 3.5 \). 2. Next, we continue to \( x=2 \): At \( x=1.5 \), \( y=f(1.5)\approx 3.5 \): \[ \frac{dy}{dx} = 1.5 + 3.5 = 5 \] The next value of \( y \) is: \[ y_2 = y_1 + f'(x_1) \cdot \Delta x = 3.5 + 5 \cdot 0.5 = 3.5 + 2.5 = 6 \] Thus, the approximation for \( f(2) \) using Euler's method is \( f(2) \approx 6 \).