Welcome to a lesson
on classifying differential equations.
There are many types of differential equations
and we classify them into different categories
based on their properties.
Let us go over the basic classifications.
Let's begin with the distinction
between ordinary and partial differential equations.
Ordinary differential equations, or ODEs,
are equations where the derivatives are taken
with respect to only one variable.
That is there's only one independent variable.
And below we have three examples.
We have the exponential growth equation,
Newton's law of cooling
and the mechanical vibrations equation.
Notice in all three equations
the derivatives are taken with respect to one variable,
the variable T.
Partial differential equations, or PDEs,
are equations that depend on partial derivatives
of several variables.
That is, there are several independent variables.
And again, below we have three examples.
We have the transport equation,
the heat equation and the wave equation in two dimensions.
For the transport and heat equation,
the partial derivatives are taken
with respect to two variables, the variables T and X,
and for the wave equation,
the partial derivatives are taken with respect
to three variables, the variables T, X and Y.
If there are several equations working together
or more than one,
we have a system of differential equations.
For example, Y prime equals X
and X prime equals Y is a simple system
of ordinary differential equations.
Maxwell's equations for electromagnetics,
shown below, are a system of partial differential equations.
Recall from calc three
the divergence operator and the curl operator
can be written out in partial derivatives
of functions involved in the X, Y, and Z variables.
And now let's discuss the order
of the differential equation or system.
The order is simply the order
of the largest derivative that appears.
If the highest derivative that appears
is the first derivative,
the equation is of first order.
If the highest derivative that appears
is the second derivative, then the equation
is of second order.
For example, Newton's law of cooling above
is of first order because it only involves
the first derivative,
while the mechanical vibrations equation
is a second order equation
because the highest derivative is the second derivative.
The equation governing the transversal vibration in a beam
shown below is a fourth order partial differential equation.
It is fourth order as at least one derivative
is the fourth partial derivative.
It does not matter that the derivative
with respect to T is only of second order.
The equation is still classified as fourth order.
We also distinguish how dependent variables appear
in the equation or system.
In particular, we say an equation is linear
if the dependent variable or variables
and their derivatives appear linearly.
That is, only as first powers.
They're not multiplied together
and no other functions of the dependent variables appear.
In other words, the equation is the sum of terms
where each term is some function
of the independent variables or some function
of the independent variables multiplied
by a dependent variable or it's derivative.
Otherwise, the equation is called non-linear.
For example, an ordinary differential equation is linear
if it can be written in the form shown below.
Notice the equation is the sum of terms
where each term is some function
of the independent variable X
or some function of X multiplied by the dependent variable Y
or it's derivative.
You may want to pause a video and study this form.
The functions A sub zero through A sub N
are called the coefficients
and the equation is allowed to depend arbitrarily
on the independent variable.
So the differential equation shown below, equation three,
is still a linear equation, as Y and its derivatives
only appear linearly, meaning raised to the first power.
All the equations and systems above as examples are linear.
It may not be immediately obvious for Maxwell's equations
unless we write out the divergence and curl
in terms of partial derivatives.
Let's see some non-linear equations.
For example, Burger's equation shown below
is a non-linear second order partial differential equation.
It is non-linear because Y and the partial
Y with respect to X are being multiplied together.
The equation DX DT equals X squared
is a non-linear first order differential equation,
as there is a second power of the dependent variable X.
Because we have DX DT in the equation,
we know X is the dependent variable
and T is a dependent variable.
So because of the X squared,
we have a nonlinear first order differential equation.
A linear equation may further be called homogeneous
if all terms depend on the dependent variable.
That is, if no term is a function
of the independent variables alone.
Otherwise, the equation is called nonhomogeneous
or nonhomogenous, or inhomogeneous, or inhomogeneous.
A homogeneous linear ODE
can be written in the form shown below.
Notice no term is a function
of the independent variable alone.
For some examples, the exponential growth equation,
wave equation and transport equation
all shown below on the left are homogeneous.
Again, no term is a function
of the independent variable or variables.
You may notice for the first two equations
the right side of the equation is not equal to zero,
but that doesn't matter.
As long as we have a linear equation
where no term is a function
of the independent variables alone,
we do have a homogeneous linear ODE.
In contrast, the mechanical vibrations equation below
is nonhomogeneous as long as F of T
is not the zero function.
Notice here, T is the independent variable.
Similarly, if the ambient temperature A is non-zero,
Newton's law of cooling is also nonhomogeneous.
Again, a homogeneous linear ODE
can always be written in the form shown here
in the blue box.
If we compare the form of a homogeneous linear ODE
to the form of a linear ODE,
notice for the homogeneous linear ODE,
there is no function B of X.
If the coefficients of a linear equation
are actually constant functions,
then the equation is said to have constant coefficients.
The coefficients are the functions
multiplying the dependent variable,
or one of it's derivatives,
not the function B of X standing alone.
A constant coefficient nonhomogeneous ODE
is an equation in the form shown below
where A sub zero through A sub N are all constants,
but the function B may depend on the independent variable X
which is shown here on the right.
For some examples, the mechanical vibrations equation below
is a constant coefficient nonhomogeneous
second order differential equation.
The same nomenclature applies to PDEs.
So the transport equation, heat equation
and wave equation all shown below
are examples of constant coefficient
linear partial differential equations.
And then finally, an equation or system
is called autonomous if the equation does not depend
on the independent variable.
For autonomous ordinary differential equations,
the independent variable is then thought of as time.
Autonomous equation means an equation
does not change with time.
For example, in Newton's law of cooling, again shown below,
is autonomous and so is equation four,
the equation DX DT equals X squared.
Notice both equations do not depend
on the independent variable T.
On the other hand, the mechanical vibrations equation
and equation three below are not autonomous.
The mechanical vibrations equation is not autonomous
because it does depend on the independent variable T
and equation three is also not autonomous
because it depends on the dependent variable X.
I hope you found this helpful.