Welcome to an introductory lesson on differential equations.
In this video will define a differential equation and also classify a
differential equation by its type, order, and linearity.
Differential equation is an equation that contains an unknown function and its derivatives.
One famous differential equation is Newton's law:
force equals mass times acceleration.
Now you might be asking: wait a minute, this equation does not contain any derivatives.
But remember a the acceleration
can be defined by the derivative of the velocity function or dv/dt,
or the second derivative of the position function
as we see here. So if we perform these two substitutions for a
we would have two differential equations here
with the derivative of the velocity function,
and here with the second derivative of the position function.
Let's look at some other differential equations as well.
Notice equations one through four
contain ordinary derivatives and equations five and six contain
partial derivatives. All of which are differential equations.
Which brings us to the two types of differential equations.
There are ordinary differential equations, or ODE's,
and partial differential equations called PDE's.
Ordinary differential equations contain only
ordinary derivatives with respect to one independent variable.
Here are equations one through four from the previous slide.
Notice how they only contain ordinary derivatives
with respect to x.
And partial differential equations contain partial derivatives
with respect to two or more independent variables.
And here are the equations five and six from the previous slide.
Notice how they contain partial derivatives
and therefore derivatives with respect to two or more independent variables.
Now let's talk about the order of
differential equations and this applies for both ODE's and PDE's.
The order of a differential equation is the order of the highest derivative
in the equation. So let's classify each of these differential equations
This first equation contains a first
ordinary derivative as we see here, therefore this is a first order ODE.
Notice the second equation contains two terms that has derivatives
but we're looking for the highest order derivative, which should be here,
and notice how this is a second derivative and therefore this is a second-order ODE.
In the third equation, again we're looking for the highest derivative
which should be this term here, and notice how it's a fourth derivative
and therefore this is a fourth-order ODE.
We need to be a little careful on this next equation.
Remember this three and this five here are exponents
and we're looking for the highest derivative,
which again would be this term here.
Therefore this is a second order ODE.
In the fifth example, we have a first partial derivative here and it's the only partial derivative in the equation
therefore this is a first-order PDE.
In the last equation contains a second
partial derivative here and therefore this is a
second-order PDE.
Now let's talk about linear and nonlinear differential equations.
A linear differential equation can be written in the following form:
and there are a few things you notice about this form of a differential equation.
First the dependent variable in this case y
and all of its derivatives are always to the first power.
Remember these parenthesis here represent the order of the derivative not the power.
So y and all of its derivatives are always to the first power.
Second, there are no products of the function y and its derivatives.
So notice none of these products contain y and the derivative of y.
And then lastly, the coefficients can be zero,
constants, or functions of t.
A nonlinear differential equation is a differential equation that is not linear.
Let's take a look at some examples of this.
We want to determine the degree of each differential equation and then determine if it is
linear or nonlinear.
This derivative here.
And now to see if its linear or nonlinear, we wanna see if it fits the form of this
equation here in red.
Meaning we want to make sure the dependent variable and all its derivatives
are always to the first power.
Which they are.
There are no products of the function y and its derivative.
Here we have a function of t times the derivative. As well as here.
So condition two has been met and the coefficients can be zero, constants,
or functions of t. Which is also true.
So this is also a linear differential equation.
Looking at the second equation, here's the highest order derivative
therefore this is a second-order ODE.
Now let's see if its linear.
When taking a look at these products here, notice how we
have y squared times
the second derivative of y, which violates condition two
where there are no products are the function y and its derivative.
Therefore this is a nonlinear differential equation
Looking at equation four, we have a first derivative here
raised to the second power but because this is a first derivative
this is a first-order ODE.
And let's see if its linear.
Condition one is a dependent variable
and all of its derivatives are always to the first power.
Well, this is raised to the second power therefore it's not
a linear differential equation. This is nonlinear.
And now for the last example, here's the highest derivative
and therefore this is a third-order ODE.
And now let's see if it's linear.
Again, the dependent variable and all of its derivatives are to the first power.
That's true.
There are no products of the function y and its derivatives.
Which is also true.
And the coefficients can be zero, constants and functions of t.
Which is also true.
Therefore this is a linear third-order ODE.
Okay, so hope you found this introductory video helpful. Thank you for watching.