4.1.8. Differential Equations¶
\newcommand{\dfdt}[2]{\frac{d^{#1} #2}{d t^{#1}}}
The Laplace transform is an important mathematical tool to solve differential equations. Many physical systems with input x(t) and output y(t) can be physically modelled with a differential equation of the form:
a_n \dfdt{n}{y(t)} + \cdots + a_1 \dfdt{}{y(t)} + a_0 y(t) =
b_m \dfdt{m}{x(t)} + \cdots + b_1 \dfdt{}{x(t)} + b_0 x(t)
Using the Laplace transform we can transform this equation into:
a_n s^n Y(s) + \cdots + a_1 s Y(s) + a_0 Y(s) =
b_m s^m X(s) + \cdots + a_1 s X(s) + b_0 X(s)
The transfer function of this system then equals:
H(s) = \frac{Y(s)}{X(s)} =
\frac{b_m s^m + \cdots + b_1 s + b_0}{a_n s^n + \cdots + a_1 s + a_0}