Final answer:
To derive the time-domain equation for the output in terms of the input, apply relevant differential equations to your system; in electrical engineering, this often involves solving for the behavior of capacitors, inductors, and resistors in a circuit.
Step-by-step explanation:
To derive the time-domain equation for the output in terms of the input, you would typically use differential equations that are relevant to the system you're examining. For an electrical circuit, this might mean applying Kirchhoff's laws or using equations that describe the behavior of specific circuit components such as resistors, capacitors, and inductors.
For example, in an RLC circuit, the charge on the capacitor can be described by a second-order differential equation, which may look similar to Equation 15.23 referenced in your question. By taking the first and second derivatives with respect to time and substituting them back into that equation, you can prove that a proposed solution is correct. Similarly, for a series RC circuit, you can find the time dependence of the current by solving a first-order differential equation.
Once these equations are solved, they yield time-domain expressions for voltages and currents in the circuit. For instance, the induced voltage across an inductor is given by VL(t) = -L(dI/dt), where I(t) is the current function that you would have found by integrating the governing differential equation.