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We saw in class how to use Picard's successive approximation method to approximate a solution for an ODE by "guessing" $\Phi_0$ and then improving the guess using the formula: $\Phi_{n+1}(x) = \int_{0}^{x}f[t, \Phi_n(t)]dt$ Until now I only saw it applied to simple first-order differential equations, but now I have a system of linear equations: $ \left\{\begin{matrix} \dot{x} = y\\ \dot{y} = -x - \frac{8}{5}y \end{matrix}\right. $ For which I'm supposed to calculate approximations using this method. How can I do that?

Thanks!

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    @DavidMitra Thanks! don't you want to write it as an answer so that I can accept it?2012-02-26

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Scroll down to "Extension to First Order Systems in 2D" here, where the iteration method for a system of two linear equations is described. Note that you'll need some initial values for the method.

Beware, though, there is an obvious typo in the link: in equation (8) there, the formula for $X_{n+1}(t)$ should read $X_{n+1}(t)=\color{maroon}{x_0}+\int_{t_0}^t f\bigl(s, X_n(s), Y_n(s)\bigr)\, ds$.