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Differential equations are not transform operations.
It is hard to explain you the target application of each one transform.
The main reason to apply transform to some equation is to reduce complexity level, and allow you work in a more friendly domain.
for moving average system we do have a defining equation, simply put the values of input signal sum them and divide by the number of values, then why do convolution?
Convolution im the area I work (image processing), is used to detect correlation betwen 2 image patterns.
Depending on magnitude of image matrix, convolution transform could be a faster way, comparing to Fourier transform.
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