Chapter 10 - Differential Equations - 10.3 Euler's Method - 10.3 Exercises - Page 550: 22

The 6 values for $x_{i},y_{i}$ and for $0\leq x \leq1$ are shown in the table below

We are given $\frac{dy}{dx}=x-2xy$ so that $g(x,y)=x-2xy$ Since $x=0, y=1$ $g(x_{0},y_{0})=0$ and $y_{1}=y_{0}+g(x_{0},y_{0})h=1+0\times0.2=1$ Now $x_{1}=0.1, y_{1}=1$ and $g(x_{1},y_{1})=-0.1$ Then $y_{2}=1+(-0.1)\times0.2=0.96$ $y_{3}=0.886$ $y_{4}=0.794$ $y_{5}=0.7$ $y_{6}=0.62$