Numerical methods: Euler's method

Let $x' = \sin(xt)$, and $x(0)=1$. Approximate $x(1)$ using Euler's method with step sizes 1, 0.5, 0.25. Use a calculator and compute up to 4 decimal digits.

Let $x' = 2t$, and $x(0)=0$.

1. Approximate $x(4)$ using Euler's method with step sizes 4, 2, and 1.
2. Solve exactly, and compute the errors.
3. Compute the factor by which the errors changed.

Let $x' = x e^{xt+1}$, and $x(0)=0$.

1. Approximate $x(4)$ using Euler's method with step sizes 4, 2, and 1.
2. Guess an exact solution based on part a) and compute the errors.

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.