- Coupled Differential Equations
- Input-Output Differential Equation
- Transformation to other representations

When analyzing a physical system, the first task is generally to develop a mathematical description of the system in the form of differential equations. Typically a complex system will have several differential equations. The equations are said to be "coupled" if output variables (e.g., position or voltage) appear in more than one equation.

Two examples follow, one of a mechanical system, and one of an electrical system.

System |
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Free Body Diagrams |
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Coupled Differential Equations |

The system is thus represented by two differential equations:

The equations are said to be coupled because x_{1} appears in both equation (as does x_{2}).

System |
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Sum currents at nodes(note, each arrow shows direction of current into or out of node) |
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Coupled Differential Equations |

The system is thus represented by two differential equations:

The equations are said to be coupled because e_{1} appears in both equation (as does
e_{2}).

Developing a set of coupled differential equations is typically only the first step in solving a problem with linear systems. The next step

A more useful form for describing a system is that of a single *
input-output* differential equation. In such a description terms with
the output and its derivatives goes on the left side of the equation, terms with
the input and its derivatives goes on the right.

As an example consider the two coupled equations from the mechanical system above.

If we wish to solve for x_{1}, we can simply solve
the first equation for x_{2}

and put this expression into the second equation

Simplifying, we get

By convention the differential equation is written

Although this last expression is still very complicated, it
is a single third order differential equation relating the output (x_{1})
to the input (F_{e}). Using standard techniques, this equation can
be solved in a straightforward manner.

Note: as expected all terms in front of x1 and its derivatives have the same sign. This is a general rule for passive (i.e., no motors, amplifiers...) systems.

It is often the case that a simple substitution, such as the one done above is impossible. The next example demonstrates this.

For example consider the case:

where the x_{1} and x_{2} are system
variables, y_{in} is an input and the a_{n} are all constants.
In this case, if we want a single differential equation with s1 as output and
yin as input, it is not clear how to proceed since we cannot easily solve for x2
(as we did in the previous example). What we can do in cases like this is
to replace each derivative by a multiplication by a variable "s" (you'll see why
this works when you study Laplace Transforms; for now, accept it without
proof. We can solve the resulting algebraic equation, put it in terms of
positive powers of "s." Each "s" in the final result is replaced by a
derivative.

Let's apply the technique to this example. First replace derivatives by "s"

Now we can solve the first equation for x2 and put this into the second equation

Multiply by a_{2}s+a_{3} to get positive
powers of s (no "s" terms in denominator).

Now we collect like powers of s, and write the differential equation in descending order of derivative, with the output on the left and the input on the right.

Since the differential equation is equivalent to the other mathematical representations of systems, there must be a way to transform from one representation to another. These methods are discussed here.

© Copyright 2005 to 2015 Erik Cheever This page may be freely used for educational purposes.

Erik Cheever Department of Engineering Swarthmore College