Chapter 10 - Differential Equations - 10.3 Euler's Method - 10.3 Exercises - Page 550: 20

The 6 values for $x_{i},y_{i}$and for $0\leq x \leq 1$ are shown in the table below

We are given $\frac{dy}{dx}=y$ so that $g(x,y)=y$ Since $x=0, y=1$ $g(x_{0},y_{0})=1$ and $y_{1}=y_{0}+g(x_{0},y_{0})h=1+1\times0.2=1.2$ Now $x_{1}=0.1, y_{1}=1.2$ and $g(x_{1},y_{1})=1.2$ Then $y_{2}=1.2+1.2\times0.2=1.44$ $y_{3}=1.728$ $y_{4}=2.074$ $y_{5}=2.488$ $y_{6}=2.986$