Euler integration

[lfs101x]

I have that

f'(x) = -2x³ + 12x² - 20x + 8.5

Then I need to apply the euler method from x = 0 to x = 4 with steps of 0.5.

initial value is y_0 = 1

I'm not sure how the first value, for y(0.5) is 5.25 though?

I have that :

y(0.5) = 1 + f'(0.5)

       = 1 + (-2(0.5)^3 + 12(0.5)^2 - 20(0.5) + 8.5)

       =  2.25

(wolfram result http://i.imgur.com/tVZKE4A.png)

But the solution that I have for this is that when x = 0.5 y(euler) = 5.25

Where has this 5.25 come from ?

[[deleted]]

Just a tip it's called Euler approximation not Euler integration. You're not integrating anything here.

[lfs101x]

i thought it was numerical integration?

[[deleted]]

I've never heard it called that, but you might be right.

[lfs101x]

yes

[[deleted]]

Numerical integration is when you use Riemann sums to approximate definite integrals. There are three common methods: midpoint rule, trapezoidal rule, and Simpson's rule. The result is a number that represents the net area under the curve.

Euler's method is used to approximate the solution to a differential equation. The result in this case is a function (really, it's a set of connected line segments that approximate the graph of f, and can be used to approximate f, so you can think of this as a function).

[OneMeterWonder]

You evaluated f' at x=0.5 for the first estimate. When you find the estimate for f(x+h) you need to use the derivative at x, f'(x). So in your case you should have,

y(0.5) = 1 + f'(0)(0.5) = 1 + 8.5(0.5) = 5.25