General Form of an N-th Order LCCDE:

A continuous-time LTI system can often be described by a linear constant-coefficient differential equation relating the output y(t), its derivatives, the input x(t), and its derivatives.
The general form is:
a_N * d^N y(t)/dt^N + ... + a_1 * dy(t)/dt + a_0 * y(t) = b_M * d^M x(t)/dt^M + ... + b_1 * dx(t)/dt + b_0 * x(t)
Here, a_k and b_k are constant coefficients, N is the order of the system (highest derivative of the output), and M is the highest derivative of the input.
Examples: RC circuit, RLC circuit, mass-spring-damper system.

Concept:

This solution describes the system's behavior when the input is zero (x(t) = 0). It represents the system's internal "modes" or "natural frequencies" determined by its inherent properties (e.g., resistance, inductance, capacitance in circuits; mass, spring constant, damping coefficient in mechanical systems).

Procedure:

1. Set the right-hand side of the LCCDE to zero (homogeneous equation).
2. Assume a solution of the form y_h(t) = C * e^(st) (exponential form, as exponentials are eigenfunctions of LTI systems).
3. Substitute this form into the homogeneous equation and factor out C * e^(st).
4. This yields the Characteristic Equation (or Auxiliary Equation), which is a polynomial in 's'. The roots of this polynomial are the Characteristic Roots (or Eigenvalues, or Natural Frequencies) of the system.
5. Based on the nature of the roots, construct the homogeneous solution:
6. Distinct Real Roots (s1, s2, ...): y_h(t) = C1 * e^(s1t) + C2 * e^(s2t) + ...
7. Repeated Real Roots (s1 occurring k times): y_h(t) = (C1 + C2t + ... + Ckt^(k-1)) * e^(s1*t)
8. Complex Conjugate Roots (alpha +/- jbeta): These always appear in pairs for real coefficients. Each pair contributes a term of the form e^(alphat) * (A * cos(betat) + B * sin(betat)) or C * e^(alphat) * cos(betat + phi). These represent oscillating behaviors that decay or grow depending on alpha.
9. The constants (C1, C2, A, B, etc.) are determined later using initial conditions.

Concept:

This solution describes the system's behavior directly caused by the non-zero input signal x(t). It accounts for the "forced" or "driven" part of the response.

Method of Undetermined Coefficients:

The most common approach for standard input forms. The form of the particular solution is assumed to be similar to the input signal, but possibly including its derivatives.
- If x(t) is a constant (K): Assume y_p(t) = A (a constant).
- If x(t) is an exponential (K * e^(alphat)): Assume y_p(t) = A * e^(alphat).
- If x(t) is a sinusoid (K * cos(omegat) or K * sin(omegat)): Assume y_p(t) = A * cos(omegat) + B * sin(omegat).
- If x(t) is a polynomial (K * t^n): Assume y_p(t) = A_n * t^n + ... + A_0.
- Special Case (Resonance): If the form assumed for y_p(t) is already part of the homogeneous solution (i.e., the input frequency matches a natural frequency), then the assumed form must be multiplied by 't' (or t^k if it's a repeated root). For example, if x(t) = e^(s1t) and s1 is a characteristic root, assume y_p(t) = A * t * e^(s1t).

Procedure:

Substitute the assumed form of y_p(t) and its derivatives into the original non-homogeneous differential equation and solve for the unknown coefficients (A, B, etc.) by equating coefficients of like terms on both sides.

Concept:

The complete solution to the differential equation is the sum of the homogeneous and particular solutions:
y(t) = y_h(t) + y_p(t)
The constants in y_h(t) are then determined by applying the initial conditions (values of y(0), y'(0), y''(0), etc.) to the total solution.

Natural Response:

• Definition: The response of the system that depends solely on the internal dynamics (characteristic roots) and the initial energy stored in the system (initial conditions), assuming the input is zero.
• Mathematical Link: This is precisely the homogeneous solution, y_h(t), with its constants determined by how the initial conditions "launch" the system into its natural modes.
• Behavior: For stable systems, the natural response typically decays to zero over time (transient behavior). For unstable systems, it grows unbounded.

Forced Response:

• Definition: The response of the system that is directly caused by the applied input signal, assuming all initial conditions are zero. It reflects how the system is "driven" by the external stimulus.
• Mathematical Link: This is the particular solution, y_p(t).
• Behavior: For stable systems, the forced response often persists as long as the input is present, representing the steady-state behavior (e.g., the output of an AC circuit after initial transients die down).

Importance of Initial Conditions:

To find a unique solution to an N-th order differential equation, N independent initial conditions are required (e.g., y(0), y'(0), ..., y^(N-1)(0)). These conditions describe the energy stored or the state of the system at the beginning of the analysis (often t=0).

Zero-Input Response (y_zi(t)): Response Due to Stored Energy Only

• Concept: The output of the system when the input signal x(t) is identically zero for all time, but the system has non-zero initial conditions. It's solely due to the "memory" or energy already present in the system.
• Calculation: This is the homogeneous solution y_h(t), where the constants are determined by applying the given initial conditions directly to y_h(t) and its derivatives at t=0.

Zero-State Response (y_zs(t)): Response Due to Input Only

• Concept: The output of the system when the system starts from a "zero state" (i.e., all initial conditions are zero), and a non-zero input signal x(t) is applied. It represents the system's pure response to the external stimulus.
• Calculation: This is precisely the convolution integral y_zs(t) = x(t) * h(t). To find this, you first need the impulse response h(t) of the system, which can be derived from the differential equation (e.g., by finding the solution to the differential equation with x(t) = delta(t) and zero initial conditions).

Total Response as a Sum of Components:

The total response y(t) of any LTI system is the sum of its zero-input response and its zero-state response:
y(t) = y_zi(t) + y_zs(t)

Significance:

This decomposition is powerful because it allows us to analyze the system's response to initial conditions and its response to the input independently and then simply add them. This separation simplifies complex problems and provides deeper insights into the system's behavior.