‫And so if this is your general solution here and then you have these kind of relationships here, then
‫you can rewrite our general solution like this, the error as a function of time equals and then a times
‫the oil, the no to the power of two times T.
‫And now instead of this, I'm going to write it like this callsign three times T plus I times sine three
‫times T like this.
‫And I can do that.
‫I'm just following this formula here and then the other term will be like this plus B times the oil,
‫the number to the power of two times T and now instead of this I'm going to write it like this times.
‫And then in the brackets you will have cosine three times T minus I times sine three times T like this.
‫And now I'm going to factor out the oil and number to the power of two times T and then in the brackets
‫I will have eight times cosine, three times T plus eight times I times sine three times T plus B times
‫cosine three times T and now minus B times I times sine three times T like this.
‫And so inside the brackets I can factor out the cosigns here.
‫So I'm going to put A plus B here, times callsign three times T, and now I'm going to factor out the
‫signs here as well.
‫So I'm going to have A minus B times I times sine three times T like this.
‫And now I'm going to say that this is A plus B, I'm going to just say that this entire thing is a more
‫general constant C one.
‫And then here I'm going to take this entire thing, including the imaginary eye number, and I'm going
‫to say that it's C two.
‫So just another random constant.
‫So C one and C two, they are random constants, just like A plus B.
‫So the general solution is error as a function of time, and then you have this or the number to the
‫power of two times T and then inside the brackets you will have C one times cosine, three times T and
‫then plus C, two times sine and then three times T here.
‫So that's your general solution.
‫And now to find C one and C two, let's again assume that the initial conditions are like this error
‫at time equals zero equals one meter and the error dot at time equals zero equals zero meters per second.
‫So let's take the derivative of this solution here.
‫So error dot is a function of time equals.
‫And now according to the product rule from calculus, we first take the derivative of this part.
‫So you're going to have two times the oil and number to the power of two times T and then what you have
‫in the brackets.
‫We don't change that.
‫So it's going to be like this and now we're going to leave this unchanged, the oil and no to the power
‫of two times T and we take the derivative of the functions in the brackets.
‫So you will have minus three C one times sine three times T and then plus three times C two and then
‫cosine three times T like this.
‫Here you have the threes because of the chain rule, because you also need to take the derivative inside
‫the cosine and sine parentheses.
‫And so according to our initial conditions here, error at time equals zero seconds, then all that
‫equals one meter.
‫So we're going to put one here and here.
‫Error dot time equals zero seconds, and then all that equals a zero.
‫Now, if time equals zero, then this thing here becomes one, because the oil number to the power of
‫two times zero equals the oil.
‫The number to the power of zero equals one.
‫And then this cool three time city will also be won and then signed three times T one time equals zero,
‫then it will be zero and then in the error derivative function, again, this part will be one, then
‫the cosine part will be one, then the sine part will be zero.
‫Again, this part here will be one.
‫The sign part here will be zero.
‫And then the coastline part here will be one.
‫So the error at time equals zero seconds equals.
‫And from here you can see that since this entire term will also be zero because you will be multiplying
‫C two times zero.
‫That means that you will have it like this error at time equals zero seconds equals C one, and that's
‫it.
‫That equals your one, right?
‫So here you will only have your C one here and then you equate that to one.
‫So you know that your C one equals one.
‫You have one of your constants.
‫And then your error dot at time equals zero seconds equals from the first term, you will have two times
‫see one and from the second term you will have plus three times C two.
‫And all that will equal to zero.
‫Since you already know that your C one equals one, you can just put one here.
‫So you will have two times one plus three times C, two equals zero.
‫And now to find C two, you put two on the other side of the equation sine and you divide it by three.
‫So three times C two equals minus two and well the three goes here so C two equals minus two over three.
‫So the solution for the given initial values is the following error as a function of time equals the
‫oil number to the power of two times t like here and now in the brackets you will have it like this
‫C one here.
‫That was your one.
‫So one times cosine, three times T and now your C two was minus two of a three.
‫So you will have it here minus two over three times sine and then three times T.
‫So that will be your solution for these particular initial conditions.