Let \(y=f(x)\) be the solution to the differential equation \(\dfrac{dy}{dx}=x-y\) with initial conditions \(f(1)=3\). What is the approximation for \(f(2)\) obtained by using Euler's method with two steps of equal length starting at \(x=1\) ?

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Starting at \((1,3)\), we have a slope of:

$$ \frac{dy}{dx}\Big|_{x=1, y=-3}=1-3 = -2 $$ $$ f(1.5)=f(1)+0.5(-2) = 3-1 = 2$$

At \((1.5,2)\)

$$ \frac{dy}{dx} \Big|_{x=1.5, y=2} = 1.5-2 = -0.5 $$ $$ f(2)=f(1.5)+0.5(-0.5) = 2-0.25 = 1.75 = \boxed{\frac{7}{4}}$$