Taylor Series Expansion

Example 3:
Solve the differential equation y' = x/y, y(0)=1 by Euler's method to get y(1). Use the step
lengths h = 0.1 and 0.2 and compare the results with the analytical solution (y 2 = 1 + x 2 )
Solution:
f(x, y) = x/y,
with h = 0.1
y 1 = y 0 + h f(x 0 , y 0 ) = 1.0 + 0.1*0.0/1.0 = 1.00
y 2 = y 1 + h f(x 1 , y 1 ) = 1.0 + 0.1*0.1/1.0 = 1.01
y 3 = y 2 + h f(x 2 , y 2 ) = 1.01 + 0.1*0.2/1.01 = 1.0298
y 4 = y 3 + h f(x 3 , y 3 ) = 1.0298 + 0.1*0.3/1.0298 = 1.0589
y 5 = y 4 + h f(x 4 , y 4 ) = 1.0589 + 0.1*0.4/1.0589 = 1.0967
y 6 = y 5 + h f(x 5 , y 5 ) = 1.0967 + 0.1*0.5/1.0967 = 1.1423
y 7 = y 6 + h f(x 6 , y 6 ) = 1.1423 + 0.1*0.6/1.1423 = 1.1948
y 8 = y 7 + h f(x 7 , y 7 ) = 1.1948 + 0.1*0.7/1.1948 = 1.2534
y 9 = y 8 + h f(x 8 , y 8 ) = 1.2534 + 0.1*0.8/1.2534 = 1.3172
y 10 = y 9 + h f(x 9 , y 9 ) = 1.3172 + 0.1*0.9/1.3172 = 1.3855