The Need for Some Mathematics

Abstract

In the previous chapter, mathematical models of various physical systems (and reasons for obtaining them) were given; the models were either obtained directly in the form of first- and second-order linear differential equations with constant coefficients or were brought to that form by linearisation (or perturbation) methods. In general, mathematical models of complicated systems will take the form of nth order linear differential equations, which may be written as follows

$$\begin{array}{*{20}{c}} {{a_0}\frac{{{{\text{d}}^n}x}}{{{\text{d}}{t^n}}} + {a_1}\frac{{{{\text{d}}^{n - 1}}x}}{{{\text{d}}{t^{n - 1}}}} + {a_2}\frac{{{{\text{d}}^{n - 2}}x}}{{{\text{d}}{t^{n - 2}}}} + \ldots + {a_{n - 1}}\frac{{{\text{d}}x}}{{{\text{d}}t}} + {a_n}x} \\ { = {b_0}\frac{{{{\text{d}}^m}u}}{{{\text{d}}{t^m}}} + {b_1}\frac{{{{\text{d}}^{m - 1}}u}}{{{\text{d}}{t^{m - 1}}}} + \ldots + {b_{m - 1}}\frac{{{\text{d}}u}}{{{\text{d}}t}} + {b_m}u} \end{array}$$

(3.1)

where x(t) is the output from the system; u(t) is the input to the system; d^r/dt^r is the rth differential coefficient with respect to time; n and m are indices (n ⩾ m + 1); the coefficients a _i and b _i are constant.