Let $f$ satisfy $\dfrac{dy}{dx} = xy$ with $f(0) = 1$. Using Euler's method with two steps of size $0.5$, approximate $f(1)$.

Core Concept

At $(0,1)$ the slope is $0$, so the first step keeps $y=1$; at $(0.5,1)$ the slope is $0.5(1)=0.5$, so the second step gives $1 + 0.5(0.5) = 1.25$.