Homework

1. Consider the differential equation $y'(x) = y(x)^2$, $y(0)=1$ and let $y_i = 1/(1-x_i)$ the exact values, i.e.\ the values of the solution $y(x)=1/(1-x)$ on the grid points, $i=0,1,2,\dots,k-1$. Apply $k$-step explicit and implicit Adams' methods and compare the local error on the next grid point $y(x_k)-y_k$.

2. Use the identities

$$(1+x)^{\alpha} = \sum_{n = 0}^{\infty} {\alpha \choose n} x^n, \qquad -\ln(1-x)=\sum_{n=0}^\infty \frac{x^n}{n}$$

to show that the coefficients

$$\gamma_j^{\ast} = (-1)^j \int_0^1 { -\lambda + 1 \choose j} d\lambda$$

of the implicit Adams' method satisfy the recursion

$$0 = \gamma_{m}^\ast + \frac 1 2 \gamma_{m-1}^{\ast} + \frac 1 3 \gamma_{m-2}^\ast +\cdots + \frac 1 {m+1}\gamma_0^\ast$$