Characteristic equation.

Take the characteristic equation.

You know the characteristic equation.

But this is our characteristic equation.

And this is called the characteristic equation.

So let's write down the characteristic equation.

And actually, you end up having a characteristic equation.

Or essentially, when you're trying to solve the characteristic equation?

But it's very easy to come up with the characteristic equation, right?

You just take the characteristic equation r squared minus 3r minus 4.

This, the r squared plus 5r, plus 6, is called the characteristic equation.

And watch the previous video just to see why this characteristic equation works.

And then you have to just find use the quadratic equation to find the complex roots of the characteristic equation.

If that's our differential equation that the characteristic equation of that is Ar squared plus Br plus C is equal to 0.

So our characteristic equation is r squared plus r plus 1 is equal to 0.

Let's do a couple of problems where the roots of the characteristic equation are complex.

And watch the previous video if you don't know where this characteristic equation comes from.

Even when we did a characteristic equation, we guessed what the original general solution was.

But really the meat of this problem was solving a quadratic, which was our characteristic equation.

However you want to say it, we only have one r that satisfies the characteristic equation.

Take the characteristic equation depending on how many roots it has and whether they're real or complex.

And if the roots of this characteristic equation are real let's say we have two real roots.

Notice, we had that thing that kind of looked like a characteristic equation pop up here and there.

You do that by getting the characteristic equation r squared minus 3r minus 4 is equal to 0.

The two roots of our characteristic equation are actually the same number, r is equal to minus 2.

So the first thing we do, like we've done in the last several videos, we'll get the characteristic equation.

But what this gives us, if we make that simplification, we actually get a pretty straightforward, general solution to our differential equation, where the characteristic equation has complex roots.

And so the solutions of the characteristic equation or actually, the solutions to this original equation are r is equal to negative 2 and r is equal to minus 3.

But it's not too hard to and actually if you ever forget it, solve your characteristic equation, get your complex numbers, and just substitute it right back in this equation.

So in general, as you get the characteristic equation, and your two roots are mu plus or minus oh sorry, no.

Now my question to you is, what if the characteristic equation does not have real roots, what if they are complex?

Because the characteristic equation to get that, we substituted e to the rt, and the Laplace Transform involves very similar function.

So like we've done in every one of these constant coefficient linear second order homogeneous differential equations, let's get the characteristic equation.

So if this is our original differential equation, the characteristic equation is going to be and I'll do this in a different color 4r squared minus 8r plus 3r is equal to 0.

Those are both things that we had to do when we solve an initial value problem, when we use just traditional, the characteristic equation.

I would say, well the roots of my characteristic equation are negative B plus or minus the square root of B squared minus 4AC.

Characteristic traits?

And I know it's a little bit frustrating right now, because you're like, this is such an easy one to solve using the characteristic equation.

So the two solutions of this characteristic equation ignore that, let me scratch that out in black so you know that's not like a 30 or something the two solutions of this characteristic equation are r is equal to well 1 plus 1 2 is equal to 3 2 and r is equal to 1 minus 1 2, is equal to 1 2.

And so our roots to the characteristic equation are minus 2 just dividing both by 2 minus 2 plus or minus, we could say i or 1i, right?

And if you want to see all of that over again, you might want to watch the previous video, just to see where that characteristic equation comes from.

Standard form in odd characteristic.

Now that's a good characteristic.

And here is that characteristic.

A characteristic of your generation.