Chapter 8.3, Problem 20ES

Consider the initial-value problem

$\frac{dy}{dx}=\frac{\sqrt{y}}{2},y\left(0\right)=1$

(a) Use Euler's Method with step sizes of $\Delta x=0.2,0.1,\text{ and }0.05$ to obtain three approximations of $y\left(1\right).$

(b) Find $y\left(1\right)$ exactly.