Solving two coupled first order differential equations

Question

I have the following system of coupled differential equations:

$\frac{d x(t)}{dt} = c_1 x(t) - c_2 y(t) + \gamma_1(t)$ and $\frac{d y(t)}{dt} = c_2 x(t) - c_1 y(t) + \gamma_2(t)$

eqns := 
  {x'[t] == c1 x[t] - c2 y[t] + gamma1[t],
   y'[t] == c2 x[t] - c1 y[t] + gamma2[t]}
initialvalues := {x[0] == x0, y[0] == y0}

Here, $c_1$ and $c_2$ are constants, while as $\gamma_1$ and $\gamma_2$ are unknown functions of $t$. Given the initial conditions $x(0)=x_0$ and $y(0)=y_0$, how can I obtain a solutions $x(t)$ and $y(t)$?

Answer 1

DSolve is able to give the general solution to this set of equations, for unknown $c_1$ and $c_2$, provided that you don't include the initial conditions. If you know $c_1$ and $c_2$, then you can also apply the initial conditions:

c1 = 1; c2 = 2;
eqns = {x'[t] == c1 x[t] - c2 y[t] + γ1[t], 
  y'[t] == c2 x[t] - c1 y[t] + γ2[t]}
initialvalues = {x[0] == x0, y[0] == y0}

DSolve[{eqns, initialvalues}, {x, y}, t] // Simplify

Which gives a longish expression in terms of integrals of $\gamma_1(t)$ and $\gamma_2(t)$.