Zero state response
In electrical circuit theory, the zero state response (ZSR), is the behaviour or response of a circuit with initial state of zero. The ZSR results only from the external inputs or driving functions of the circuit and not from the initial state. The total response of the circuit is the superposition of the ZSR and the ZIR, or Zero Input Response. The ZIR results only from the initial state of the circuit and not from any external drive. The ZIR is also called the natural response, and the resonant frequencies of the ZIR are called the "natural  frequencies". Given a description of a system in the s-domain, the zero-state response can be described as Y(s)=Init(s)/a(s) where a(s) and Init(s) are system-specific. Zero state response and zero input response in integrator and differentiator circuits. One example of zero state response being used is in integrator and differentiator circuits. By examining a simple integrator circuit it can be demonstrated that when a function is put into a linear time-invariant (LTI) system, an output can be characterized by a superposition or sum of the Zero Input Response and the zero state response. A system can be  represented as formula_1 formula_2 with the input formula_3 on the left and the output formula_4 on the right. The output formula_4 can be separated into a zero input and a zero state solution with formula_6 The contributions of formula_7 and formula_1 to output formula_9 are additive and each contribution formula_7 and formula_11 vanishes with vanishing formula_7 and formula_13 This behavior constitutes a linear system. A linear system has an output that is a sum of distinct zero-input and zero-state components, each varying linearly, with the initial state of the system and the input of the system respectively. The  zero input response and zero state response are independent of each other and therefore each component can be computed independently of the other. Zero state response in integrator and differentiator circuits. The Zero State Response formula_11 represents the system output formula_9 when formula_16 When there is no influence from internal voltages or currents due to previously charged components formula_17 Zero state response varies with the system input and under zero-state conditions we could say that two independent inputs results in two independent outputs: formula_18formula_19 and formula_20formula_21 Because of linearity we can then apply the principles of superposition to achieve  formula_22formula_23 Verifications of zero state response in integrator and differentiator circuits. To arrive at general equation. The circuit to the right acts as a simple integrator circuit and will be used to verify the equation formula_24 as the zero state response of an integrator circuit. Capacitors have the current-voltage relation formula_25 where "C" is the capacitance, measured in farads, of the capacitor. By manipulating the above equation the capacitor can be shown to effectively integrate the current through it. The resulting equation also demonstrates the zero state and zero input responses to the integrator circuit. First, by integrating both  sides of the above equation formula_26 Second, by integrating the right side formula_27 Third, distribute and subtract formula_28 to get formula_29 Fourth, divide by formula_30 to achieve formula_31 By substituting formula_32 for formula_33 and formula_34 for formula_35 and by using the dummy variable formula_36 as the variable of integration the general equation formula_37 is found. To arrive at circuit specific example. The general equation can then be used to further demonstrate this verification by using the conditions of the simple integrator circuit above. By using the capacitance of 1 farad as shown in the integrator circuit above formula_38 which is  the equation containing the zero input and zero state response seen at the top of the page. To verify zero state linearity. To verify its zero state linearity set the voltage around the capacitor at time 0 equal to 0, or formula_39, meaning that there is no initial voltage. This eliminates the first term forming the equation formula_40. In accordance with the methods of linear time-invariant systems, by putting two different inputs into the integrator circuit, formula_41 and formula_42, the two different outputs formula_43 and formula_44 are found respectively. By using the superposition principle the inputs formula_41 and formula_42  can be combined to get a new input formula_47 and a new output formula_48 By integrating the right side of formula_49 formula_50 is found, which implies the system is linear at zero state, formula_39. This entire verification example could also have been done with a voltage source in place of the current source and an inductor in place of the capacitor. We would have then been solving for a current instead of a voltage. Zero state response industry uses. The circuit analysis method of breaking a system output down into a zero state and zero input response is used industry  wide including circuits, control systems, signal processing, and electromagnetics. Also most circuit simulation software, such as SPICE, support the method in one form or another.