Problem 6 - Euler's Method

Answer the following questions concerning Euler's method for the ordinary differential equation $\frac{dy}{dx} = f(x,y)$. $x$ is discretized with the equal spacing as $x_{i+1} = x_i + h\;(h>0)$ and the approximation of $y(x_i)$ is denoted by $y_i$.

1.

Expression $y_{i+1}$ in terms of $y_i$. Assuming $y_i$ has no error, estimate the error in $y_{i+1}$.

2.

The differential equation $\frac{dy}{dx}=-\lambda y\;(\hbox{Re}(\lambda)>0)$, $y(0)=1$ is solved by Euler's method in the interval $[0,1]$ with $N$ equal spacings. Estimate the error at $x=1$ and show what power of $N$ the leading form of the error is proportional to as $N\rightarrow\infty$.

3.

Let one solve the differential equation in the previous question in the open interval $[0,\infty)$. Show the region of $\lambda$ on the complex plane where the solution by Euler's method satisfies $\lim_{i\rightarrow\infty} y_i=0$.