‫Welcome back.
‫Let's now perform the solution check, so this is our solution for this differential equation, while
‫having the initial conditions like this error at time equals zero seconds equals one meter and then
‫error dot the time.
‫Derivative of the error at time equals zero seconds equals zero meters per second.
‫So let's not take the derivative of this solution.
‫This derivative looks like this.
‫First of all, you take the derivative of this part and you leave the other part alone and then you
‫do it in reverse.
‫The oil, the number to the power of two times, you leave it alone and then you take the derivative
‫of what you have in the brackets here.
‫And so here you have three times two over three.
‫You can cancel out the threes, then you can rewrite this entire thing like this where you multiply
‫two by what you have here in this bracket, and then you will have here minus four over three.
‫And here you will have two.
‫And you can see that two times cosine, three times t can be cancelled out with this minus two times
‫cosine three times T, right.
‫So this one and this one, they will be canceled out.
‫And so your error dot equals minus thirteen over three times the oil number to the power of two times
‫T and then times sine and then you have three times T here.
‫So that's your solution derivative then this will be the second derivative of your error and now you're
‫going to put all these things into your differential equation here.
‫This will be your second derivative here.
‫So e double that that you put here, then this here, this one will be your first derivative, which
‫is this one here that you're going to put right here.
‫And then you have minus four times this error that so you see you have here minus four times the error
‫dot here and then this one here.
‫This will be your 13 times error.
‫Your error solution is this one here.
‫And so you can see that you can cancel out this term with this term here because they amount to zero.
‫And so if you put it all together, then you will have minus twenty six over three plus fifty two over
‫three and then minus twenty six over three and all that multiplied by the oil, a number to the power
‫of two times T times sine and then three T, so minus twenty six over three comes from here and then
‫here you will have fifty to over three.
‫If you multiply minus four times minus thirteen over three then you will get fifty two of three and
‫then here you will have minus 13 times two over three and you will get minus twenty six over three.
‫And then you will factor out this oil number with its power and sine three times t you will factor them
‫out and you can see that here in this bracket you will have zero.
‫So if you multiply zero by all this here, then you will get zero.
‫And look, this part here will be zero.
‫And then on this side of the equation, sine you will also have zero.
‫So zero equals zero and therefore the solution is correct.
‫Alright then in the next video, we're going to go back to our controller context.
‫Thank you very much.