The mathematical tool used to describe first-order circuits.

• A. First-Order Differential Equation
• B. Second-Order Differential Equation
• C. Algebraic Equation
• D. Integral Equation

Answer: (A)

Describing a first-order circuit requires a first-order differential equation because there is a single energy storage element (a capacitor or an inductor) whose stored energy makes the circuit’s state evolve in time. The governing relationship is set by how that stored quantity changes with time and interacts with the resistive elements.

Take a simple RC circuit with an input source. The capacitor voltage is the state variable, and the current through the capacitor is i_C = C dv_C/dt. The resistor current is i_R = (v_in − v_C)/R. Since the same current flows, i_C = i_R, which leads to C dv_C/dt = (v_in − v_C)/R. Rearranging gives a first-order differential equation: dv_C/dt + (1/RC) v_C = (1/RC) v_in. This equation fully describes how the circuit’s voltage evolves over time in response to the input.

Similarly, an RL circuit yields a first-order differential equation: L di/dt + R i = v_in, which describes how the current changes with time due to a single energy storage element (the inductor).

Why not the other options? A second-order differential equation would come from having two energy storage elements (like a resistor, inductor, and capacitor combination that leads to two dynamic states). An algebraic equation describes a static, time-invariant situation with no dynamics. An integral equation isn’t the standard starting point for modeling the time evolution of a basic first-order circuit, though its solutions may involve integrals; the fundamental description remains a first-order differential equation.