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‫Welcome back.

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‫So now you have learned how to go from London one and London two to the Constance Kay one and Kate two,

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‫and now let's look at the same thing.

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‫But from a different point of view, suppose that again, this is your differential equation, this

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‫one right here with your key one and key to values.

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‫The thing with differential equations is that they are often used to represent systems.

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‫That means that they could also be converted into state space equations.

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‫And also note that I have been using the word LTI or linear time invariant a lot in front of our differential

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‫equations.

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‫Well, let's put our differential equation in this space form.

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‫I can rewrite this equation like this error dot equals error dot and then error double dot equals K

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‫one times error plus key two times error dot.

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‫And it's a valid statement, right?

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‫There is nothing wrong with that.

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‫In the second equation, I simply put this term and this term to the other side of the equation, sine

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‫and well error dot equals zero dot.

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‫It's just true, right.

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‫It's like five equals five.

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‫But of course you might be asking why would I write error dot equals error dot.

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‫What's the purpose of it?

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‫Well, the reason why I wrote it like this is because now I can represent this system in a vector matrix

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‫form.

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‫I can have a vector here so I can have error dot here then error double dot here, then I have some

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‫kind of matrix here and then I have another vector which is error and then error dot.

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‫You see it has two elements here, error and error dot.

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‫So thanks to this error that equals error dot equation, I can now fill in the first row of this matrix.

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‫So if error that equals error dot, then that means that you will have zero here and you will have one

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‫here.

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‫And now if you look at the second equation here, this one.

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‫Then you can see that you will have a key one here and key to right over here.

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‫So tell me now, does it seem familiar to you?

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‫Well, it does to me, because to me what it looks like is this x dot as a vector equals and then a

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‫matrix times the state vector.

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‫So this is the state time derivative vector equals eight times the state vector.

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‫So this matrix here, that's your A matrix.

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‫And so if you assume that your key values K one and key to our constants, then this system here in

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‫fact is a linear time invariant or an LTI system.

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‫So you see I've taken this differential equation and I have converted it into an LTI states based system,

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‫and that is why I have been using the word linear time invariant in front of our differential equation.

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‫I have always been saying homogeneous, linear time invariant or LTI second order differential equation.

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‫And the way I know that this differential equation is an LTI differential equation is because if I transform

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‫it into a state based form, then I will have an LTI state based form in which this a matrix is a constant

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‫matrix.

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‫It's time invariant.

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‫It doesn't change with time.

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‫And another thing that you might have noticed is that you only have an A matrix here.

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‫There is no B matrix here.

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‫That's because this differential equation here is homogeneous.

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‫Let's take a look at a non homogeneous differential equation.

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‫So this is our differential equation here and this one is not a homogeneous one because it has a term

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‫here on the other side of the equation sine that is not zero.

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‫And so as an exercise, try to put this differential equation into your state based form, try doing

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‫it yourself and then we'll look at it in the next video.

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‫Thank you very much.