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I am trying to understand why I should use Runge-Kutta methods. What I understood is that it is used to calculate a derivative at some point. But my question then is why not differentiate and plug in the numbers directly? Thank you

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Could you give a concrete example of what you mean? –  lhf May 17 '12 at 2:42
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up vote 3 down vote accepted

Runge-Kutta is not used to calculate a derivative, it is used to solve an initial value problem for a differential equation. For example: you want to find $y(10)$ given the differential equation $\frac{dy}{dx} = \sin(y + x)$ and the initial value $y(0)=1$.

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