Definitions and Terminology (Level 2) In this video we are going to practice classifying
differential equations. Let's first classify ordinary differential equations.
State the order of the given Ordinary Differential Equation and determine whether the equation
is linear or nonlinear.
the quantity (1-x) times y double prime minus 4x times y prime plus 5y equals cos(x), where y = f(x).
Alright first thing to notice is that we are
told that y is a function of x. This means that the independent variable is x and the
dependent variable is y. The first thing you always need to do is to identify the independent
variable and dependent variable before you even attempt to classify the differential
equation. The next thing I want you to notice is that this ODE is written using prime notation,
we can also rewrite this ODE using Leibnitz notation as follows. Recall that the advantage
of using this notation is that it explicitly shows the independent variable x and the dependent
variable y. In any case you can use either form as long as you keep track which variable
represents the dependent and independent variable. Alright let's first classify the order of the
ODE, recall that the order is the highest derivative of the equation, in this ODE we
have two derivatives a second derivative and a first derivate, the highest of these two
would be the second derivate so we classify this ODE as second order.
Next, we need to identify if the ODE is linear or nonlinear, in general the procedure in
determining if an ODE is linear in form is as follows, we first check to see if the dependent
variable y and its derivatives are linear in form meaning are they raised to the power
of 1, looking at the dependent variable y and its derivatives we see that they are all
raised to the power of 1, if they were raised to the power of a number other than 1 like
2 or negative 3, then this ODE would’ve been Nonlinear. Next we need to check if the
products of the dependent variable y and its derivatives are solely in terms of the independent
variable x, looking at the product of the second derivative we see that it’s the quantity
(1-x), which is solely in terms of the independent variable x, in addition the product of the
first derivative is also in terms of the independent variable x, and lastly the dependent variable
y contains the constant 5, if any of these expressions contained the dependent variable
y or its derivatives then this ODE would've been nonlinear, lastly, we need to check for transcendental
functions such as trigonometric functions, exponential functions and logarithmic functions
and make sure they contain only the independent variable x, in this ODE we see that we have
a cosine of x term the argument contains the independent variable x, if it contained the
dependent variable y then this ODE would've been classified as a Nonlinear ODE. Since all three
criteria are met we conclude that this ODE is Linear in form. We can classify this equation
as a second order linear ODE.
Notice that transcendental functions are technically non linear functions, but keep in mind that
the determining factor for an ODE to be linear is if the transcendental functions solely
contain the independent variable in the argument. This is why you need to identify the dependent
and independent variable first before you attempt to classify an ODE.
Alright let’s try the next example.
x times the third derivative of y with respect to x minus the first derivative of y with respect to x raised to the power of 4 plus y equals 0.
Alright this time we are given an ODE using
Leibniz notation, if you like you can rewrite this ODE using prime notation as follows.
Let's first determine the order of the ODE, scanning the ODE we see that we have a third
derivative and a first derivative, the highest of these two is the third derivative so this
ODE is a third order ODE. Next let’s check to see if this ODE is linear; let’s first
check to make sure that the dependent variable and its derivative are linear in form.
The third derivative is raised to the power of 1 so this derivative is linear in form,
looking at the first derivative we see that it’s being raised to the power of 4, this automatically
makes this ODE Nonlinear so we don’t need to go through the rest of steps to determine
if the ODE is linear. So this equation can be classified as a third order Nonlinear ODE.
A common mistake that many students make is to use the exponent in this case 4 to classify
the order of the ODE remember it’s the highest derivative and not the exponent that determines
the order of the ODE, so make sure you don’t make this common mistake. Alright let’s
try the next example.
The quantity (1-y^2) times the second derivative of y with respect to t plus t times dy/dt plus y equals e^t.
Notice that this ODE has the variable t as
the independent variable usually when we have time as the independent variable we can rewrite
the differential equation by using dot notation also known as Newton’s dot notation as follows.
This is an alternative way of writing this ODE that has time as the independent variable.
Alright let's first identify the order of the ODE, notice that we have a second derivative
and a first derivative, the highest one of these is the second derivative so this ODE
is a second order ODE, next let’s check to see if this ODE is linear, we start by
making sure that the dependent variable y and its derivatives are linear in form, for
this ODE we see that this is the case, so now we go ahead and check if the products
of the dependent variable and its derivatives are solely in terms of the independent variable
t, for the second derivative notice that the product is the quantity one minus y squared,
this quantity contains the dependent variable y which in turn is being multiplied by the
second derivative. This automatically makes the ODE non linear. In the end this equation
can be classified as a 2nd order Nonlinear ODE. It’s non linear because the product
of the second derivative contains an expression in terms of the dependent variable y.
Alright let’s try the final example.
The 4th derivative of y plus 2 times y prime plus e raised to the power of y equals x. Where y = f(x).
Once again we have an ODE written using prime
notation; we can also rewrite this ODE by using Leibniz notation as follows.
Notice that when using prime notation only the first three derivatives are labeled by using primes
any higher order derivatives are labeled by using a superscript enclosed in parenthesis
like this 4th derivative, this is why it’s sometimes convenient to use Leibniz notation
when dealing with ODE’s that contain derivatives that are greater than three.
On that note let’s classify the order of this ODE, notice that we have a 4th derivative
and a first derivative don’t confuse the 4 as an exponent if it were an exponent we
would not enclose the 4 with parenthesis. The greater of these two derivatives is the
4th derivative so this ODE is a 4th order ODE. Next let’s check and see if this ODE
is linear. First we need to check that the dependent variable y and its derivatives are
linear in form, inspecting the ODE we see that this is the case, next we need to check
that the products of the dependent variable y and its derivatives are solely expressed
in terms of the independent variable x, once again inspecting the ODE we see that this
is the case, the final step is to check if any transcendental functions are written solely
in terms of the independent variable x, notice that this ODE contains the transcendental
function e raised to the power of y, this function is in terms of the dependent variable
y, this makes the ODE nonlinear. If this function was written in terms of the independent variable
x then we would’ve had a linear ODE. So in the end we can classify this equation as
a 4th order nonlinear ODE, it’s nonlinear because of the e to the y term.
Alright so this is the basic procedure you want to follow when classifying ODE’s once
you get more practice you will immediately be able to classify the order of an ODE and
determine whether it’s linear or nonlinear in a matter of seconds. In our next video
we will go over a couple more challenging examples.