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I am new in the field of systems and signals, and I have a rather basic for the majority of the group, question:

Can we find the impulse response function of homogeneous ODE, instead of its zero-input response?

for example, we have the following 2nd order homogeneous ODE:

$a_{2}\ddot{x}+a_{1}\dot{x}+a_{0}{x} = 0$

, where the output is the $x(t)$ given the initial conditions

I understand that if it were: $a_{2}\ddot{x}+a_{1}\dot{x}+a_{0}{x} = f(t)$

the $f(t)$ would be its input, $x(t)$ its output given the input, and we could find the impulse response by replacing the $f(t)$ with $\delta (t)$.

Now that the input is zero, how can we find what the output would look like with respect to any input?

Is this:

$a_{2}\ddot{x}+a_{1}\dot{x}+a_{0}{x} = \delta (t)$

even allowed, for an initially homogeneous equation?

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  • $\begingroup$ Alex, formally speaking a homogeneous ODE with initial conditions will not constitude an LTI system and won't admit an impulse response: a response which can be used through convolution to compute outputs for arbitrary admitted inputs... $\endgroup$ – Fat32 Oct 18 '19 at 20:47
  • $\begingroup$ I don't understand this question. The response of a system described by an ODE with constant parameters is the sum of the zero-input (natural) response and the zero-state (forced) response (which assumes zero initial conditions, i.e., zero initial state). $\endgroup$ – Rodrigo de Azevedo Oct 19 '19 at 6:50
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The answer to your question

Can we find the impulse response function of homogeneous ODE, instead of its zero-input response?

is "no", because only a system with an input and an output can have an impulse response, a homogeneous ODE doesn't have an impulse response.

The impulse response of a system, possibly described by an ODE, is the zero-state response to an input signal $x(t)=\delta(t)$.

Of course, a system can also have a zero-input response, which is obtained by solving the corresponding homogeneous ODE with the appropriate initial conditions, but this response has nothing to do with the system's impulse response.

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