# Transformation: Single Diff Eq ↔ Transfer Function

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## Single Differential Equation to Transfer Function

If a system is represented by a single nth order differential equation, it is easy to represent it in transfer function form.  Starting with a third order differential equation with x(t) as input and y(t) as output.

To find the transfer function, first take the Laplace Transform of the differential equation (with zero initial conditions).  Recall that differentiation in the time domain is equivalent to multiplication by "s" in the Laplace domain.

The transfer function is then the ratio of output to input and is often called H(s).

Note: This notation takes increasing subscripts for the an and bn coefficients as the power of s (or order of derivative decreases) while some references use decreasing subscripts with decreasing power.  This notation was chosen here in part because it is consistent with MatLab's use.

For the general case of an nth order differential equation with m derivatives of the input (superscripted numbers in parentheses indicate the order of the derivative):

This can be written in even more compact notation:

## Transfer Function to Single Differential Equation

Going from a transfer function to a single nth order differential equation is equally straightforward; the procedure is simply reversed.  Starting with a third order transfer function with x(t) as input and y(t) as output.

To find the transfer function, first write an equation for X(s) and Y(s), and then take the inverse Laplace Transform.  Recall that multiplication by "s" in the Laplace domain is equivalent to differentiation in the time domain.

For the general case of an nth order transfer function:

This can be written in even more compact notation:

## Example: Transforming Between Single Differential Equation and Transfer Function

##### Example: Single Differential Equation to Transfer Function

Consider the system shown with fa(t) as input and x(t) as output.

Find the transfer function relating x(t) to fa(t).

Solution: Take the Laplace Transform of both equations with zero initial conditions (so derivatives in time are replaced by multiplications by "s" in the Laplace domain).

Now solve for the ration of X(s) to Fa(s) (i.e, the ration of output to input).  This is the transfer function.

##### Example: Transfer Function to Single Differential Equation

Find the differential equation that represents the system with transfer function:

Solution: Separate the equation so that the output terms, X(s), are on the left and the input terms, Fa(s), are on the right.  Make sure there are only positive powers of s.

Now take the inverse Laplace Transform (so multiplications by "s" in the Laplace domain are replaced by derivatives in time).

References