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.

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

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
.

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

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