Let \( y=f(x) \) be the solution to the differential equation \( \frac{d y}{d x}=2 x+y \) with initial condition \( f(1)=0 \). What is the approximation for \( f(2) \) obtained by using Euler's method with two steps of equal length, starting at \( x=1 \)

To begin with Euler's method, we need to determine the step size based on the interval from \( x=1 \) to \( x=2 \), which has a length of 1. Since we want to use two steps of equal length, the step size \( h \) will be: \[ h = \frac{2 - 1}{2} = 0.5 \] Next, we will use the initial condition \( f(1) = 0 \) to approximate \( f(2) \). 1. **First Step:** \[ x_0 = 1, \quad f(x_0) = f(1) = 0 \] The slope at this point (using the differential equation) is: \[ \frac{d y}{d x} = 2(1) + 0 = 2 \] We can update our value of \( f \): \[ f(1.5) \approx f(1) + h \cdot \frac{d y}{d x} = 0 + 0.5 \cdot 2 = 1 \] 2. **Second Step:** Now we will calculate the slope at the new point \( x_1 = 1.5 \) with \( f(1.5) = 1 \): \[ x_1 = 1.5, \quad f(x_1) = 1 \] Using the differential equation, we find the new slope: \[ \frac{d y}{d x} = 2(1.5) + 1 = 3 + 1 = 4 \] Update the value of \( f \) again: \[ f(2) \approx f(1.5) + h \cdot \frac{d y}{d x} = 1 + 0.5 \cdot 4 = 1 + 2 = 3 \] Thus, the approximation for \( f(2) \) using Euler's method with two steps of equal length is: \[ \boxed{3} \]