###### Abstract
The second method requires knowledge of the system's impulse response function and is expressed by the following equations:  
\[W(j\omega,t)=\mathcal{F}\{h(t,\tau)\}=\int\limits_{0}^{t}h(t,\tau)\mathrm{e}^ {-\mathrm{i}\omega\tau}\mathrm{d}\tau \tag{3}\]  
\[W(s,t)=\mathcal{L}\{h(t,\tau)\}=\int\limits_{0}^{t}h(t,\tau)\mathrm{e}^{-s \tau}\mathrm{d}\tau \tag{4}\]  
From a mathematical perspective, Formulas (3) and (4) define the classical Fourier and Laplace transforms of the impulse response with respect to the variable \(\tau\). The main difference between the time-frequency analysis of LTV systems and that of stationary systems is the inability to proceed directly from differential equations to transfer functions. In this conventional approach, the determination of the impulse response function or the system response to an arbitrary input is of paramount importance in determining the frequency characteristics of LTV systems. The main challenge is to find fundamental solutions to the homogeneous parametric differential equations satisfying Equation (1). Analytical solutions to differential equations are available only in specific cases, which heavily depend on the equation's order and the form of the parametric functions. The fundamental solutions of these equations, even for first or second order, are highly intricate and can be represented by special functions from mathematical physics, such as Bessel functions of the first and second kind with non-integral orders, as well as confluent hypergeometric functions [16]. Assuming periodic variation of the parameters of LPTV systems (with the constraint that they are at most second-order systems), well-known equations from mathematical physics, such as the Mathieu, Meissner, or Hill equations, can be used. However, the key issue remains the stability of these solutions. Depending on the choice of the coefficient variation over time, the systems can show instabilities. Furthermore, in most cases, the solutions to these equations are not available in closed form (analytically), requiring support from numerical computations in the analysis. While the exact analytical determination of this function is feasible in a limited number of cases, even then, the function is represented by special functions, rendering further analysis exceedingly challenging. The authors emphasize that while the theory of differential equations provides fundamental solutions (in some cases) through integral equations, the determination of closed-form solutions--such as the response of the system to a non-zero external forcing or an impulse response function--is often analytically intractable. Consequently, these solutions typically require the use of analytical numerical methods, and one must often rely on approximate solutions to effectively address the problem. Given these complexities, the Frequency Symbolic Method (FSM), introduced by the authors in their earlier works [15] and briefly summarized in Section 2, becomes a valuable tool for describing and analyzing LPTV systems in the frequency domain, as it enables the frequency description of a specific class of parametric systems with periodically varying parameters in conjunction with the L.A. [PERSON] equation [1]. The FSM allows direct formulation and analysis in the frequency domain, eliminating the need to solve time-domain differential equations. This approach is particularly advantageous as it avoids the intermediate step of time-domain analysis, which can be both complex and computationally expensive for systems with periodically varying parameters (LPTV). The FSM is applied using the [PERSON]. [PERSON] matrix equation, which provides a more organized and symbolic representation of LPTV systems. This matrix-based approach decomposes the problem into several independent matrix equations, making analysis easier and allowing a more methodical approach to solving the system.