Exercise 104

Consider $\dfrac{dy}{dx} = x+y$, $y(0)=1$.

1. Approximate $y(1)$ using the improved Euler's method (see above) with step sizes $h=\nicefrac{1}{4}$ and $h=\nicefrac{1}{8}$.
2. Approximate $y(1)$ using Euler's method with $h=\nicefrac{1}{4}$ and $h=\nicefrac{1}{8}$.
3. Solve exactly, find the exact value of $y(1)$.
4. Compute the errors, and the factors by which the errors changed.