Determine if system is linear, time-invariant and causal from differential equation 
Determine if the system described by the differential equation
$$\ddot{y}(t) + 2\dot{y}(t)-6y(t) = 2x(t) $$
is linear, time-invariant and causal. $y(t)$ is the system output and $x(t)$ is the system input.

A system is linear if the input/output terms are linear - meaning there are no $y^2(t)$ or something like that. So the system is definitely linear.
A system is time-invariant if the coefficients of the differential equation are constants. So the system is definitely time-invariant.
A system is causal if the system's current output only depends on past and present inputs.
My question: Is it possible to determine from the differential if a system is causal/non-causal?
 A: I used Mathematic to solve the system:
$$
  y(t) =  k e^{\lambda_1 t} \left( c_1+
\int_1^t e^{-\lambda_1 \xi} x(\xi) d\xi
\right)
-  ke^{\lambda_2 t} \left(c_2 +
\int_1^t e^{-\lambda_2 \xi} x(\xi) d\xi\right), \tag{*}
$$
where $\lambda_1 = -1+\sqrt{7}, \lambda_2 = -1-\sqrt{7}, k = \frac{1}{2 \sqrt{7}}$.
If $y(t)$ is bounded as $t\to \infty$ for any choice of $c_1,c_2$, then the system is causal, i.e. we can choose $c_1,c_2$ to meet the initial conditions and $y(t)$ only depends on the past values $x(s),s\leq t$. However, if at lest one of $c_1$ and $c_2$ has to be chosen so that the system does not "explode", then the future values of $x(s),s>t$ determine the current state. (Imagine it as preparing a dam for a future rain to prevent the dam braking when the rain season comes.)
Now looking at equation (*) we can see that the effect of $c_2$ diminishes over time as $\lambda_2<0$, but $c_1$ has to be put
$$
  c_1 = - \int_1^t e^{-a \xi} x(\xi) d\xi
$$
so that $y(t)$ does not explode as $t\to \infty$, because $\lambda_1>0$.
In general, as system is causal when both of the roots $\lambda_1,\lambda_2$ of its characteristic equation have strictly negative real parts; and it is non-cause when at least one of them has strictly positive real part.
A: Yes, it is possible to determine if a system is causal from the differential equation that describes it.
A system is causal if and only if the impulse response of the system, which is the response of the system to a delta function input, is zero for all time prior to the time at which the delta function is applied. In other words, the output of a causal system at any given time depends only on the input at that time and earlier, and not on the input at any future times.
If the differential equation for the system has no terms that depend on future values of the input or output, then the system is causal.
In the given case of the differential equation in the question,
$$\ddot{y}(t) + 2\dot{y}(t)-6y(t) = 2x(t),$$
there are no terms that depend on future values of the input or output, so the system described by this differential equation is causal.
