Chapter 10 - Differential Equations - 10.3 Euler's Method - 10.3 Exercises - Page 550: 8

$y(1.5)=2.783$

We are given $\frac{dy}{dx}=e^{-y}+e^{x}$ so that $g(x,y)=e^{-y}+e^{x}$ Since $x=1, y=1$ $g(x_{0},y_{0})=e^{-1}+e^{1}\approx3.086$ and $y_{1}=y_{0}+g(x_{0},y_{0})h=1+3.086\times0.1=1.3086$ Now $x_{1}=1.1, y_{1}=1.3086$ and $g(x_{1},y_{1})=3.274$ Then $y_{2}=1.3086+(3.274)\times0.1=1.636$ $y_{3}=1.636+(3.515)\times0.1=1.9875$ $y_{4}=1.9875+(3.806)\times0.1=2.368$ $y_{5}=2.368+(4.149)\times0.1=2.783$ $y_{6}=2.783+(4.544)\times0.1=3.237$