Which of the following statements are highlighted as characteristics of linear, time-invariant systems?

• A. The integral of the input equals the integral of the output.
• B. The derivative of the input equals the derivative of the output.
• C. Neither of the above.
• D. Both of the above.

Answer: (D)

The key idea here is what defines linear, time-invariant systems: linearity and time invariance. Linearity means the system response scales with the input and sums with input: if you double the input, the output doubles; if you add two inputs, the outputs add. Time invariance means shifting the input in time produces the same shifted output. These properties do not by themselves impose that the integral of the input must equal the integral of the output, nor that the derivative of the input must equal the derivative of the output. Such equalities only hold for very specific systems: an integrator (where the output is the integral of the input) or a differentiator (where the output is the derivative of the input), but these are particular cases, not general features of all LTI systems. Therefore, neither of the stated equalities is a general characteristic of linear, time-invariant systems; the correct view is that neither holds in general.