# How to illustrate the transfer function with a given equation?

Homework/Revision question:

Define the transfer function of a linear system. Illustrate your answer by considering the system governed by the equation:

$\frac{dy}{dt}+ay=bx$

where x and y are functions of t, and a and b are constants.

Now I know that Transfer functions, also known as System functions is an equation that demonstrates the relationship between the spectral content of an input, and spectral content of an output. It can be written as $Y(w) = H(w)X(w)$ where Y(w) is the output and is the Fourier Transformation of y(t). X(w) is the input and is the Fourier Transformation of x(t). And I can do calculations when those are given.

But I don't quite understand what the question means by asking me to illustrate the definition while considering the provided equation?

Any help would be much appreciated.

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It means compute $H(\omega)$ explicitly for the given system. –  copper.hat Jun 20 '12 at 7:15
@copper.hat how would i do that? –  Synia Jun 20 '12 at 7:40
Let $\hat{x}, \hat{y}$ be the transforms of $x,y$. Note that the transform of $\frac{d y}{d t}$ is $i \omega \hat{y}$. Now take the transform of both sides of the differential equation and solve for $\hat{y}$. The form of $H(\omega)$ should be apparent then. –  copper.hat Jun 20 '12 at 14:43