Chapter 10 - Differential Equations - 10.3 Euler's Method - 10.3 Exercises - Page 550: 6

$y(1.4)=0.584$

We are given $\frac{dy}{dx}=1+\frac{y}{x}$ so that $g(x,y)=1+\frac{y}{x}$ Since $x=1, y=0$ $g(x_{0},y_{0})=1$ and $y_{1}=y_{0}+g(x_{0},y_{0})h=0+1\times0.1=0.1$ Now $x_{1}=1.1, y_{1}=0.1$ and $g(x_{1},y_{1})=1.09$ Then $y_{2}=0.1+(1.09)\times0.1=0.209$ $y_{3}=0.209+(1.174)\times0.1=0.326$ $y_{4}=0.326+(1.251)\times0.1=0.451$ $y_{5}=0.451+(1.322)\times0.1=0.584$ $y(1.4)= y_{5}=0.584$