Numerical Analysis

Exercise 1.8

1. Draw the directional field and solution curve through $(1,0),(0,1),(1,1)$ of the followings:
(a) $y^{\prime} = y$
(b) $y^{\prime} -xy + 1 = 0$
2. Using Euler's method to approximate the followings:
(a) $y^{\prime} = x + y, y(0) = 0$. Find $y(3)$. Here, $h = 1$
(b) $y^{\prime} = x^{2}, y(1) = 2$. Find $y(4)$, Here, $h = \frac{1}{2}$

Answer
1(a), (b) omitted

2. (a) $$\begin{array}{ll} x_{0} = 0 & y_{0} = 0\\ x_{1} = 1 & y_{1} ... ... x_{3} = 3 & y_{3} = f(2,1) + y_{2} = 3+ 1 = 4 \end{array} \ \$$

(b) $$\begin{array}{ll} x_{0} = 1 & y_{0} = 2\\ x_{1} = 3/2 & y_{1... ... = 4 & y_{6} = f(7/2,106/8)(1/2) + 106/8 = 155/8 \end{array} \$$