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# 11.2. Applying the Runge-Kutta Method to Second-Order Initial Value Problems

## Problem

You need to numerically solve a second-order differential equation of the form:

## Solution

This is a standard initial value problem, and you can implement any of a number of standard numerical integration techniques to solve it using Excel and VBA.

## Discussion

Consider this equation and initial conditions:

Physically, this equation represents the equation of motion for an object moving under some applied thrust, T.[*] Here, m represents the mass of the object, C d represents a drag factor, and s represents the displacement (position) of the object. This equation allows us to compute the motion of this object.

To solve this equation, it helps to rewrite it in the form of two first-order equations. Let v = ds/dt represent the velocity of the object. Now we have the following:

These equations represent ...

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