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:

\[H(s) = \frac{Y(s)}{X(s)} = \frac{b_m s^m + \cdots + b_1 s + b_0}{a_n s^n + \cdots + a_1 s + a_0}\]