Taylor Series Expansion

Example 2:
Use Eulers method to solve for y[0.1] from y' = x + y + xy, y(0) = 1 with h = 0.01 also
estimate how small h would need to obtain four-decimal accuracy.
Solution : f(x, y) = x + y + xy,
y 1 = y 0 + h f(x 0 , y 0 ) = 1.0 + .01*(0 + 1 + 0*1) = 1.01
y 2 = y 1 + h f(x 1 , y 1 ) = 1.01 + .01*(0.01 + 1.01 + 0.01*1.01) =1.02
y 3 = y 2 + h f(x 2 , y 2 ) = 1.02 + .01*(0.02 + 1.02 + 0.02*1.02) =1.031
y 4 = y 3 + h f(x 3 , y 3 ) = 1.031 + .01*(0.03 + 1.031 + 0.03*1.031) =1.042
y 5 = y 4 + h f(x 4 , y 4 ) = 1.042 + .01*(0.04 + 1.042 + 0.04*1.042) = 1.053
y 6 = y 5 + h f(x 5 , y 5 ) = 1.053 + .01*(0.05 + 1.053 + 0.05*1.053) = 1.065
y 7 = y 6 + h f(x 6 , y 6 ) = 1.065 + .01*(0.06 + 1.065 + 0.06*1.065) = 1.076
y 8 = y 8 + h f(x 7 , y 7 ) = 1.076 + .01*(0.07 + 1.076 + 0.07*1.076) = 1.089
y 9 = y 9 + h f(x 8 , y 8 ) = 1.089 + .01*(0.08 + 1.089 + 0.08*1.089) = 1.101
y 10 = y 10 + h f(x 9 , y 9 ) = 1.101 + .01*(0.09 + 1.101 + 0.09*1.101) = 1.114