<s>[INST] Solve the system of differential equations and plot the curves given the initial conditions. We are given the following system of ordinary differential equations:
$$\dot{x} = x - 4y \quad \dot{y} = x - 2y -4.$$
Thus, the slope of the trajectories is given by
$$\frac{dy}{dx} = \frac{\dot{y}}{\dot{x}} = \frac{x - 2y -4}{x-4y}.$$
We have the initial conditions $x(0) = 15$ and $y(0) = 5$.
How do I solve this system? The book does not give the solution but claims that we can solve for the trajectories. It does not suggest a method. It also asks what happens for other initial conditions.
I tried treating each separate equation as a linear equation and solved them via integrating factors, but that of course resulted in an integral equation on the right-hand side. I was not sure if that was the right way to go about it.
 [/INST] Mathematica:
$$\left\{x(t)\to \frac{1}{7} e^{-t/2} \left(56 e^{t/2} \sin ^2\left(\frac{\sqrt{7}
   t}{2}\right)-3 \sqrt{7} \sin \left(\frac{\sqrt{7} t}{2}\right)+56 e^{t/2} \cos^2\left(\frac{\sqrt{7} t}{2}\right)+49 \cos \left(\frac{\sqrt{7}
   t}{2}\right)\right), y(t) \to \frac{1}{7} e^{-t/2} \left(14 e^{t/2} \sin
   ^2\left(\frac{\sqrt{7} t}{2}\right)+5 \sqrt{7} \sin \left(\frac{\sqrt{7}
   t}{2}\right)+14 e^{t/2} \cos ^2\left(\frac{\sqrt{7} t}{2}\right)+21 \cos
   \left(\frac{\sqrt{7} t}{2}\right)\right)\right\}$$

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