‫Welcome back.
‫So let's now check our solutions.
‫This is your solution for this differential equation.
‫And remember, these concerns are three or five and two or five, because your error at time equals
‫zero equals one.
‫And then the time derivative of your error at time equals zero equals zero.
‫So the error was in meters in our case, because we have X, Y and Z dimensions and they are measured
‫in meters.
‫And then the error that was meters per second because you have a time derivative here.
‫And so if we take the derivative of the error of this general solution, then you will have it like
‫this three over five times two, and then the oil and number to the power of two times T plus and then
‫two over five and then times minus three.
‫And then the oil, the number to the power of minus three times.
‫In other words, it equals six over five times the oil.
‫Number to the power of two times Steet and then minus six over five times the oil number to the power
‫of minus three times T and well, the second time derivative of this general solution will be like this.
‫So six over five here, times two and then again times the oil number to the power of two times T and
‫then minus six over five like here times minus three times the oil or no to the power of minus three
‫times T and so you're going to have 12 over five.
‫And then the oil, the number to its power, and then plus 18 over five times the oil, the number,
‫and again, it's another power and now this thing here will go here, then this thing here will go here.
‫And then this thing here that will go here.
‫That would be the second time derivative for this one here.
‫That would be the first time derivative for this one here and then minus six times and then the expression
‫for the error itself.
‫And so if you write it all out, then you will have it in this form here.
‫And then you can factor out the oil number to the power of two times T and then in the brackets you
‫will have this thing.
‫And then you can also factor out the oil, a number to the power of minus three times T, and this is
‫what you will have in the brackets.
‫And so that entire thing has to equal to zero because this differential equation here equals to zero.
‫So on the right side of the equation side, you have a zero here and it's the same zero like here.
‫So this entire thing has to equal zero.
‫And it does because in the brackets here, this equals zero and also this equals zero.
‫So the left side of the equation sine will also be zero.
‫And so zero equals zero.
‫And therefore your solution, which is this one here, is correct.
‫And so that's how you check whether your solution is correct or not.
‫You take this function, you take the first derivative, you take the second derivative, and you put
‫all that inside your original differential equation.
‫And you see if the number on one side of the equation sine is the same number, like on the other side
‫of the equation sine.
‫And that's it for that.
‫But now I'm going to give you another exercise.
‫If you remember, our very original differential equation was error double date minus.
‫OK, two times ever that minus K, one times error equals zero.
‫And let's say that now your key values are like this, your key one equals minus 13.
‫All right, so this key, one value, that's this one here, and then your key to value equals four.
‫So this key to value, that's this one here.
‫And so your differential equation now is erodable dot minus and then K2 is four.
‫So four times error dot and then minus and then K one is minus 13.
‫So you will end up having plus 13 here times.
‫Error equals zero.
‫And your exercise now is to solve this differential equation.
‫Start doing it just like we did with the previous differential equation.
‫And don't worry, you will see in the end how it all comes together in our controller.
‫I'm just laying the groundwork for that.
‫So try to solve this differential equation and I'll see you in the next video.
‫Thank you very much.