impulse response VS zero-input response
Basically, you can solve this kind of EDO using two different techniques (which also have c different meaning).

Classical Method - you will need the full EDO, containing input and outputs with initial conditions applied over the whole solution y(t), where y(t) = yhomogeneous(t) + yforced(t). The first term, you solve the homogeneous EDO considering NO CONDITIONS. Then, following the pattern of input you exactly find the forced contribution. Then, you add both contributions and apply the initial conditions...
In this technique you totally lose the feeling about the system itself.

zero state response and zero input response. In this case, the whole solution is written as y(t) = y0(t) (zero input response) + x(t)h(t)  (zero state response, as convolution of both signals. The text cutof the asterics).
To find y0(t) you do not consider the input x(t), and apply condition at t= 0^-. It happens because it is completely independent of the input. The system responds just because of the storage energy.
The second term, zero state response, it is the convolution of input and impulse response of the system. So, you need to find the impulse response of EDO. It is done considering the whole EDO , e.g., D4y -D3y +2D2y -2Dy +y = Dx -3x . First you have to solve the homogeneous equation and apply some special conditions generalized... y=Dy=D2y=0 and D3y=1. Lets call it yn(t).... Then you apply the input operator over this yn(t). It means,
D (yn(t)) -3yn(t)... After that, you finally have h(t)...
So, you do the convolution of h(t) and x(t) and find the zero state response
y