The impulse response is the system's reaction to an input impulse.

In an RC circuit, the voltage source is the input, and the capacitor's voltage is the output.

The system's state and output response before and after the input excitation are distinctly defined.

Kirchhoff's law forms an input signal equation, with the capacitor's current and voltage providing the output.

Substitution of the expression for current and division by RC yields a differential equation, with the impulse signal's output as the impulse response.

The time constant is introduced, and the differential equation is multiplied by the integrating factor on both sides. It is simplified using the impulse function's sampling property.

Both sides of the equation are further simplified and integrated within the system's limits, resulting in the equation that includes a step function and a dummy integration variable,τ. This equation is solved to calculate the RC circuit's impulse response.

The graph shows an instant jump in capacitor voltage at a time equal to zero, a contradiction due to the unrealizable nature of a pure input impulse.