Tuesday, 1 October 2013

Approximation using Euler's method.

Approximation using Euler's method.

Consider the initial value problem $$\dfrac{dy}{dx} = y,y(0) =1$$
Approximate $y(1)$ using Euler's method with a step size of
$\dfrac{1}{n}$, where $n$ is an arbitrary natural number. Use this
approximation to write Euler's number $e$ as a limit of an expression in
$n$. How large do you have to choose $n$ in order to approximate $e$ up to
an error of at most 0.1? Comment on the quality of approximate in this
example.
What I did is the following:
$y(1) \approx y_1 = y_0 +hf(x_0,y_0) = 1+hf(0,1) = 1+\dfrac{1}{n}$
This is where I stuck, am I on the right direction? What should I do next?

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