Let $y=f(x)$ be the solution to the differential equation $\dfrac{dy}{dx}=x-y-1$ with the initial condition $f(1)=-2$. What is the approximation for $f(1.4)$ if Euler's method is used, starting at $x=1$ with two steps of equal size?

Summary Submit Skip Question

Starting at $(1,-2)$, we have a slope of:

$$\frac{dy}{dx}\Big|_{x=1, y=-2}=1-(-2)-1 = 2$$ $$f(1.2)=f(1)+0.2(2) = -2+0.4 = -1.6$$

At $(1.2,-1.6)$

$$\frac{dy}{dx} \Big|_{x=1.2, y=-1.6} = 1.2-(-1.6)-1 = 1.8$$ $$f(1.4)=f(1.2)+0.2(1.8) = -1.6+0.36 = \boxed{-1.24}$$