Answer
$x_{ 0} = 0 $, $ y_{ 0} =2 $
$x_{1 } = .1$, $ y_{ 1} = 2.2 $
$x_{2 } = .2 $, $ y_{2 } = 2.43$
$x_{ 3} = .3 $, $ y_{ 3} = 2.693 $
$x_{4 } = .4 $, $ y_{4 } = 2.9923 $
$x_{5 } = .5 $, $ y_{ 5} = 3.33153$
$x_{ 6} = .6 $, $ y_{6} = 3.7147 $
$x_{ 7} =.7 $, $ y_{ 7} = 4.1462 $
$x_{8 } =.8 $, $ y_{8 } = 4.6308$
$x_{9 } = .9 $, $ y_{9 } = 5.1739 $
$x_{10} = .10$, $y_{10} = 5.7813$
Work Step by Step
$ y'= x+y $, (0,2) , n=10, h=0.1
Use Eulers method in the form
$x_1= x_0 + h$, $ y_1= y_0 +hF(x_0, y_0) $
Start with
$x_1= 0.1$ and $ y_1= 2 + .1(2) = 2.2$
$x_{2 } = .2 $, $ y_{2 } =2.2 + .1(2.3)= 2.43$
$x_{ 3} = .3 $, $ y_{ 3} =2.43+0.1(2.63)= 2.693 $
$x_{4 } = .4 $, $ y_{4 } =2.693+ 0.1(2.993)= 2.9923 $
$x_{5 } = .5 $, $ y_{ 5} = 2.9923+ 0.1(3.3923)= 3.33153$
$x_{ 6} = .6 $, $ y_{6} =3.3315+ 0.1(3.83)= 3.7147 $
$x_{ 7} =.7 $, $ y_{ 7} =3.7147 + 0.1(4.314)= 4.1462 $
$x_{8 } =.8 $, $ y_{8 } = 4.1462+ 0.1(4.8461)= 4.6308$
$x_{9 } = .9 $, $ y_{9 } = 4.6308+ 0.1(5.4307)= 5.1739 $
$x_{10} = .10$, $y_{10} = 5.1739 + 0.1(6.0738) = 5.7813$
