How do I stabilize my oscillating system?
First, combine your two variable set of first order differential equations into a single variable second order one.

$$ \frac{d^2y}{dt^2} = c \frac{dx}{dt} = acy + bc $$

$$ \frac{d^2y}{dt^2} - acy = bc $$

Solve:

$$ y(t) = C_1 e^{\sqrt{ac} \cdot t} + C_2 e^{-\sqrt{ac} \cdot t} - \frac{b}{a} $$

$$ x(t) = \int{ ( ay(t) + b ) dt }  $$

$$ x(t) = C_1 \frac{a}{\sqrt{ac}} e^{\sqrt{ac} \cdot t} - C_2 \frac{a}{\sqrt{ac}} e^{-\sqrt{ac} \cdot t} + C_3 $$

Now analyze:

Initial conditions have to be set so that $ C_1 = 0 $ or your system will grow exponentially (assuming $ ac &gt; 0 $).  Also, $ C_3 $ will need to be zero.  If $ ac &lt; 0 $ then the system will oscillate forever since there is no damping term in the second order equation.

$$ \frac{dy}{dt} = \sqrt{ac} \left( C_1 e^{\sqrt{ac} \cdot t} - C_2 e^{-\sqrt{ac} \cdot t} \right) $$

$$ \frac{dx}{dt} = a \left( C_1 e^{\sqrt{ac} \cdot t} + C_2 e^{-\sqrt{ac} \cdot t} \right) $$

Initial Conditions:

$$ y(0) = C_1 + C_2 - \frac{b}{a} $$

$$ \frac{dy}{dt}(0) = \sqrt{ac} \left( C_1 - C_2 \right) $$

$$ x(0) = \frac{a}{\sqrt{ac}} ( C_1  - C_2 ) + C_3 $$

$$ \frac{dx}{dt}(0) = a \left( C_1 + C_2 \right) $$

With $ C_1, C_3 = 0 $:

$$ y(0) = C_2 - \frac{b}{a} $$

$$ \frac{dy}{dt}(0) = - \sqrt{ac} \cdot  C_2  $$

$$ x(0) = -\frac{a}{\sqrt{ac}} \cdot C_2 $$

$$ \frac{dx}{dt}(0) = a  \cdot  C_2 $$

It's been a while, I hope I did it right.

Ced

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Followup:

Yes, you can include an $x$ term in the first equation, or a $y$ term in the second equation, or both.  Then repeat the process I went through above to put your system in this form:

$$ \ddot y + A \dot y + B y = 0 $$

Whatever constant you get on the right side will be part of the particular solution.  Add it in later as I did above.

This is the basics of solving a homogenous second degree linear differential equation:

Assume the solution is of the form $ C e^{rt} $

$$ r^2 C e^{rt} + A r C e^{rt} + B C e^{rt} = 0 $$

$$ \left( r^2  + A r + B \right) \left( C e^{rt} \right) = 0 $$

One, or both, of the factors has to be zero.  $ C e^{rt} $ won't be, so:

$$ r^2  + A r + B = 0 $$

This is a standard quadratic equation.  With real valued coefficients you will either get two real roots, or a conjugate pair of complex roots.

$$ r = - \frac{ A }{2} \pm \sqrt{ \left(\frac{ A }{2} \right)^2 - B } $$

This will give you $r_1$ and $r_2$.  Since it is a linear differential equation, any linear combination of the two solutions will also be a solution.

$$ y(t) = C_1 e^{r_1 t} +  C_2 e^{r_2 t} $$

The real part of the root will determine the growth behavior.  If it is negative the system will converge to an equilibrium point, if it is positive, the system will grow exponentially.  The imaginary part, if there, will determine the frequency of the oscillations.  Therefore, you can calculate the parameters from observations by measuring the rate of decay to get the real part(s), and measuring the frequency to get the imaginary part.  Once you have those, you should be able to solve for your $a,b,c$ and $d,e$ if you include them.

Of course, you could leave your equation set in matrix form and solve it with eigenvalues and eigenvectors, but that is a whole 'nuther discussion.

Ced