# Picard successive approximations for a system of linear differential equations

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!

• Scroll down to "Extension to First Order Systems in 2D" here. (I haven't checked it carefully, but at first glance, it looks like it has an obvious typo) – David Mitra Feb 25 '12 at 20:06
• Note that this is the problem $\dot{X} =AX$ where $X=(x,y)$ and $AX=A(x,y)=(y,-x-8/5 y)$. – checkmath Feb 25 '12 at 21:14
• @DavidMitra Thanks! don't you want to write it as an answer so that I can accept it? – Hila Feb 26 '12 at 14:12

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 $X_{n+1}(t)=\color{maroon}{x_0}+\int_{t_0}^t f\bigl(s, X_n(s), Y_n(s)\bigr)\, ds$.