# impulse response VS zero-input response

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?

• 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... – Fat32 Oct 18 '19 at 20:47
• 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). – Rodrigo de Azevedo Oct 19 '19 at 6:50

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