In linear system theory, it is easy to find the particular solution to differential equation by means of convolving the system's impulse response with the forcing function. My question is why can we not implement a similar technique to non-linear systems?
I understand that the impulse response of the non-linear system will not be proportional to the magnitude of the input (non-linear systems don't satisfy homogeneity), thus there does not exist a single impulse response that characterizes the entire system. But would it not be possible to 'pre-determine' a variety of impulse responses that correspond to input magnitudes, and then add up these responses depending on the inputs at various times to determine the overall output at a given time instance?
I feel like what I just said attempts to use super position methods to solve non-linear systems, but why exactly does superposition or additivity not hold in this sense when the inputs are applied at different time instances?