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# 11.3. Tackling Coupled Equations

## Problem

You've seen how to apply both the Runge-Kutta method and Euler's basic method using VBA for first- and second-order differential equations, but now you need to solve a system of coupled equations . You're wondering how to extend what you've learned so far to deal with this more complicated problem.

## Solution

You can apply the same VBA techniques discussed earlier with some modifications to deal with multiple equations.

## Discussion

Consider this system of coupled nonlinear differential equations with initial conditions:

Pretty, huh? This problem originally consisted of two coupled second-order equations that were reduced to four first-order equations using the same technique discussed in Recipe 11.2. This is a common technique for reducing the order of differential equations, making them more amenable to solving. This particular problem was adapted from a problem that appears in An Introduction to Linear and Nonlinear Finite Element Analysis, by Prem K. Kythe and Dongming Wei (Birkhauser). The problem represents a finite element solution of the nonlinear vibrations of a metal rod. Since this is a nonlinear problem, it requires a numerical solution. In their book, Kythe and Wei present some Matlab code to solve this problem using Euler's method .

### Euler's method

I implemented Euler's method in VBA to illustrate how to solve this same system in Excel. ...

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