# Initial rest condition for the linear constant-coefficient differential equations

Suppose that system has the input/output relation as follows

$$\sum_{k=0}^{N}a_k \frac{d^ky(t)}{dt^k} = \sum_{k=0}^{M}b_k \frac{d^kx(t)}{dt^k}$$

Where $$a_k, b_k \in \mathbb{R}$$.

Obviously we need additional conditions to determine completely the input-output relationship for the system and those auxiliary conditions specify whether the system is LTI and causal or not.

The definition of initial rest condition for a system in general is that if $$x(t) = 0$$ for $$t\lt t_0$$ then $$y(t) = 0$$ for $$t\lt t_0$$.

First of all is it true that the initial rest condition for the all systems implies $$y(t_0) = 0$$? Also I have seen expressions like $$y({t_0}^+)$$ and $$y({t_0}^-)$$ but I don't know how they relate to $$y(t_0)$$. Another question is that whether the initial rest condition is necessary and sufficient for the mentioned system to be causal and LTI? What's the rigorous proof, then? I searched on the Internet but didn't found a proof or a clear explanation.

• Please refer to Allen oppenheim's book Signal and Systems, all these points are clearly explained there. – Dsp guy sam Apr 18 at 6:45
• @Dspguysam I've seen that but it doesn't contain a proof. – S.H.W Apr 18 at 6:47
• it is there, i have specified in the answer, please have a look – Dsp guy sam Apr 18 at 7:42

For $$y(t_o^+)$$ and $$y(t_o^-)$$ relating to $$y(t_o)$$ post the quote you are looking for clarifications on, it depends on multiple context, generally that would be specifying an initial condition of a one sided Fourier transform or one sided sequence.