3. Differential Equations

When a linear system, with concentrated parameters and invariant over time, is excited with an input signal \( x\left( t \right) \), the classic mathematical model for determining the output (or response) \( y\left( t \right) \) is an n order ordinary linear differential equation with constant coefficients. In case the system is an electrical circuit, the order n is generally at most equal to the sum of the number of capacitances and inductances of the circuit, computing these numbers, however, only after capacitances and inductances in series and / or parallel have been replaced by their respective equivalents. This chapter presents a fundamental review about the theory and solution of ordinary differential equations. Moreover, it presents a systematic method to solve ordinary differential equations in which \( x\left( t \right) \) is a singular and / or causal function. This method allows to determine the initial conditions at \( t = 0_{ + } \), from the respective initial conditions at \( t = 0_{ - } \) and the differential equation itself, without any physical reasoning about the given circuit. Therefore, a good understanding of this chapter is an indispensable condition for the good follow-up of the following chapters.