Linear 1st order coupled DEs

[marc1uk]

This has got me really stumped. I've started out with an equation for the differential of a 2x2 matrix:

$$\frac{d[N]}{dz} = f(z)[E][D] - \{[W],[N]\} - c\sigma_1[N] - d\sigma_2[N]$$

where all terms in [] are matrices, {} denotes anti-commutator, and the $$\sigma$$'s are the Pauli matrices. Everything but c and d are functions of z, but they're all known functions - only N is unknown.
I know physically that this should describe equations of motion for the matrix N in which the off-diagonal terms are exponentially damped. I think it's sensible to split the equation into equations for the diagonal and off-diagonal elements. If I'm right this results in equations of the form:

$$\frac{dN_{\alpha\alpha}}{dz} = f(z)(DE)_{\alpha\alpha} - W_{\alpha\beta}N_{\beta\alpha} - W_{\beta\alpha}N_{\alpha\beta} - 2W_{\alpha\alpha}N_{\alpha\alpha}$$

and

$$\frac{dN_{\alpha\beta}}{dz} = f(z)(DE)_{\alpha\beta} - Tr[N]W_{\alpha\beta} - Tr[W]N_{\alpha\beta} - cN_{\alpha\beta} - d(\sigma_2)_{\alpha\beta}N_{\alpha\beta}$$

where $$\alpha \neq \beta$$. These are what I have to try to solve. And I'm stumped. I'm hoping someone can give me tips on solving coupled differential equations, stuff of the form:
$$\frac{dy}{dz} = f(z) + g(z)y + h(z)x$$

$$\frac{dx}{dz} = u(z) + v(z)x + w(z)y$$

Anyone got any ideas on this kind of thing? Any kind of help would be really appreciated.

(of course i suppose I'm really trying to solve:
$$\frac{dy_1}{dz} + a(z)b(y) = c(z) + d(z)u(x)$$
$$\frac{dy_2}{dz} + e(z)f(y) = g(z) + h(z)v(x)$$
$$\frac{dx}{dz} + p(z)q(x) = r(z) + s(z)t(y)$$
:uhh:

 

[gtoguy97]

)The best way to approach this type of problem is to use the method of variation of parameters. This involves writing down a general solution for the system of equations, and then finding the constants of integration by plugging in the initial conditions. You can find more information on this technique here: https://en.wikipedia.org/wiki/Variation_of_parameters.