I have two textbooks: one on differential equations (A first Course in Differential Equations, Dennis G. Zill) and one on signals and systems (Oppenheim).

The first book defines a linear constant-coefficient differential equation as having the form: $$a_0y + a_1y' + ... + a_ny^{(n)} = f(t)$$

On the other hand, my signals and systems book defines a linear constant-coefficient differential equation representing a system with input $x(t)$ and output $y(t)$ as: $$\sum_{k=0}^N a_k \frac{d^ky}{dt^k} = \sum_{k=0}^M b_k \frac{d^kx}{dt^k}$$

Now, I understand that these are two different mathematical objects: In the "standard" constant-coefficient linear differential equation, the function of interest is really just $y(t)$, as all of the coefficients as well as $f(t)$ are usually provided. On the other hand, in signals and systems, there are two functions of interest, the input $x(t)$ and the output $y(t)$.

At the same time, I feel that if I was told to "modify the definition of a linear differential constant-coefficient equation so that it represents a system", I would have instead come up with this:

$$a_0y + a_1y' + ... + a_ny^{(n)} = f(x(t))$$

My question is: Why does the second textbook instead use their more restrictive definition? What is gained by making sure $f(x)$ is actually a linear constant-coefficient expression of our input $x(t)$? What am I "losing", if I use the definition I expected instead?



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