Problem 6 - Euler's method

Answer the following questions concerning an intial value problem of the ordinary differential equation

$$\frac{dy}{dx} = f(x,y), \quad y(x_0)=y_0$$

We assume that the $f(x,y)$ has enough differentiability.

1.

The Taylor expansion of the solution $y(x)$ is given as follows:

$$y(x+h)=y(x)+hy'(x)+\frac{h^2}{2}y''(x)+\frac{h^3}{6}y'''(x)+\cdots$$

Give the expression of $y'(x)$,$y''(x)$, and $y'''(x)$ in terms of $f(x,y)$ and its partial derivatives.

2.

We define $x_{i+1} = x_i + h$ $(h>0)$. In Euler's method, $y_{i+1}$, an approximation of $y(x_{i+1})$, is given by

$$y_{i+1} = y_i + hf(x_i,y_i).$$

Assume that the $y_i$ on the righthand side is accurate and estimate the error of $y_{i+1}$.

3.

In the case of

$$f(x,y)=-\lambda y,$$

show the region of $\lambda$ for which $\lim_{i\rightarrow\infty} y_i=0$ on the complex plane of $\lambda$.