euler's methods- numerical solutions of differential equations

[liv]

can anyone explain this? i don't get why it is used.

and this may be a simple question, but how do you do:
using euler's method, approximate y(2) with i) h=0.5 and ii)0.2 for dy/dx=x/(sqrt(x^2+8), y(1)=3

[Yitzi_K]

Euler's Method: y_n+1 = y_n + h*f'(x_n)

So in this case, with h=0.5, y₂ = 3 + 0.5*1/3

That gives y₂ = 19/6, repeat that with the new values to find y₃, which is an approximation for y(2).

It's the same with steps of 0.2, it will just take 5 applications of Euler's method, each building on the last, until you've found y₆.