[MUSIC] Okay, let's consider the differential equation number three again okay, so Y double prime plus is PXY prime plus 2XY is equal to zero right? It's a normal phone right, standard phone right, and which means that, we are assuming that, X0 is a regular singular point of that differential equation okay? First we'd like to remind what it means by saying that X0 is a regular singular point okay? It means the following, okay, first, it must be a singular point which means that, either little p of X or little q of X is not analytic, it's not analytic, analytic at point X0. And the regular singular point means, furthermore, both X minus X0 p of X and X minus X0 square and q of X are analytic at X0 okay? That's the meaning of saying that, the point X0 is the regular singular point of the differential equation and with a very simple example coming from the oil differential equation, right? IX0 is a regular singular point, the user set a solution of the policy it is a for, may not work right, it may work in some lucky cases but not in general okay? So we are looking for some other method will be handling such a differential equation okay? So such a differential equation or around the point X0 which is a regular singular point of the differential equation okay? To get an idea, multiply the given equation by X-X0 squared then what simply we get, X-X0 squared, Y double prime plus X -X0 okay, then x-x0 times p of x right, do you agree? y prime plus right, then, x minus x zero squared times the q of x and that is equal to zero, it's simply by multiplying the whole equation by x minus x0 squared right, okay. And look at this then a little bit modify the differential equation, okay, you may say that the coefficient of this differential equation is a product of all the time coefficients times functions of each of you're analytic point x0. By which I mean, look at this coefficient first, x minus 6q, then look at the x-x0, then look at this one there's nothing, you may interpret it as a one time or something right, look at those three coefficients, it's kind of x square, y double prime, xy prime and the constant. Let me see, I I forgot to put the y down there okay, by those three coefficients, I'm calling it to be here, all the time coefficients okay? If both px and qx are identically one, then this is a simply oil differential equation around the singular point, regular singular point tax zero but we have something more than that right, we have more, something like this one and something like that one right? What are the characteristics of these two due coefficients, okay, x minus x0 times the p of x, x minus x0 squared times the q of x, by the definition of the regular singularity. Both of them are analytic right, this is analytic at point x0 and that one is also analytic, at point x0 right? That's the key point okay, what does that suggest to us, we might guess that here's the guess, they might take just a solution cite a solution forced all the type solution, that master looks like x-x0 to the certain power r right? Because of those two on a radical efficiency in addition times some kind of the power series okay, so that is a summation from c of n, x-x0 is over to the end right? Or you can write it if you want that, this is n0 to infinity of 6 of n x minus x02d and plus the r okay. That's our guess okay, and this guess is quite reasonable because the coefficients of differential equation is a product of all the type coefficients times functions which are analytic at point x0. So we are guessing that there might think it's just the cd solution of type of this form okay, be careful, it's not a power as it is in general unless Ali is a kind of some non negative integer okay good, so this this is right. This is, let me say okay, which is not a power as it is okay, in general, some lucky cases it might reduce to power as it is but not in general that's what I'm saying okay? And it's called, the solution of providence type for other news type solution okay, that's the key point of the theory that we are going to develop from now on. Think about it, okay let's make that claim okay, our guests to be a little bit more precise, okay, what I mean is Okay, using this one, let me have some space here. For this standard form of the differential equation, assuming that there is a regular singular point text and not, okay? In other words satisfying those two conditions, okay, so that we can rely to the differential equation into that form. Okay, now here's the claim. Okay, here's a theorem. That is called the Frobenius Theorem, okay? Okay then, okay? There is at least one, okay, one solution, okay, of Frobenius theorem. Now we know what it means, right? Then this solution is equal to y is equal to the sum of 0 to infinity of C sub n (x- x0) to the n + r, right, okay? There is a solution at least one solution of the Frobenius theorem, right? For any of this type of the infinite is, okay, we are also interested in there the range of convergence, right? What can you say about that? Okay, which converges At least four. Okay, the distance between X -X0 is creator than zero. Greater than zero and less than R where R is the distance from that point X0 to the nearest singular point. Okay, nearest singular point, okay, of the differential equation, okay, out of the next zero, right. If we are lucky then, there might be two linearly independent solution of the Frobenius theorem and we might have okay, this Frobenius theorem solution may convert this for, okay, all that satisfying. Okay, the distance between X-X0 is greater than 0 and the less than plus infinity or even. Okay, this type of thing we call it as one sided neighborhood, okay, one sided neighborhood of that point X0, okay? Because, okay, if we solved this the inequality for absolute value then it means simply for X between X0 and X0+R, okay? So we might record it as a one side of the neighborhood of X0, right? That's the monumental theorem of the proven news, okay? Okay, instead of giving that the precise proof, we'll explain the theorem through the couple of the examples, okay? We are just honestly following, following the guess our guess, right? You say that there is at least one solution of the problem is type converges in, okay, one sided neighborhood of the regular singular point, right? So plugging this Frobenius top solution into the differential equation and see what happens. Okay, differential equation is 0 = (X- X0) cubed + (X- X0) and(X-X0) Little p of x. Why? No, I forgot. Okay, here's the y double prime I need. Okay, here's a y double prime + and y prime + and X -X0 and the k and the cube X y and that is equal to 0. But that's the given differential equation, right, okay? Okay, because I'm assuming that by the regular singularity. This part, right, this is analytic at X0, right? That is also analytic at point X0, right? What does it mean by saying that those two functions on a litigate point X0? That means we can expand, right? These two functions as a power series, as a tailor series about center X0, right? So this you can write it as a sum of 0 to infinity of P sub n, okay, and X- X0 to the n, right? Then you can write it 0 to infinity of q sub n (X- X0) to the n, right? That's still an elasticity, right? Which converges in some neighborhood of X0, right? Now plugging this candidate, okay, this candidate into the differential equation then. Okay, look at the equation. One side is 0 must be equal to (X- X0) cubed y double prime. Okay, what is the y double prime from this one? Right, that's simple. Summation combined differentiation one time derivative means and plus l times c sub n times X- X0 to the n + r- 1, right? One more derivative means, right, I think you will get (n+r)(n+r-1) times C sub n (X- X0) to the n + r- 2, right. Okay, summation must start from the 0 to infinity. Multiply this one by X -X0 square. Okay, so that in fact this will be X- X0 to the n plus r, right, are you following me? Right, let's think about it. For the second part, okay, + what I thinking about the X- X0 Little times y prime. What is y prime again? By the term, by term differentiation, that is the summation. Term by term differentiation, okay, n + r C sub n (X- X0) to the n + r- 1, summation starts from 0 to infinity. Then times X- X0, that makes the power to be in plus r again, right? This part is X- X0 times y prime. Right, times this one, okay. That would be better to write it in It's the Taylor series way, okay? And this one times summations 0 to infinity of P sub n (X- X0) to the n, right?. Finally, okay? This, right, the last term. That's simply the Taylor series expansion summation q sub n. Say, this is summation 0 to infinity q of n x- x0 to the n times y, that is the summation 0 to infinity of 6 of n x- x0 to the n + r, okay? That must be a quarter 0, right, okay? What's the lowest possible power of x- x0? The lowest possible power of x- x0 is x- x0 to the r y, okay? Those are times that you can get in all those summations when n is equal to 0, right? So in particular, that means that let me say it this way, okay? Which means, okay, all coefficients of x- x0 to the n + r for or and greater than equal to 0 must be equal to 0 way, okay? That's identity for this series, infinite series, okay? In particular, Choose the lowest possible power of x- x0, and that happened when in each summation, when n is equal to 0, okay? So, okay, for n is equal to 0, okay, what do we get? When n is equal to 0, we get from the first r times r- 1 times z of 0, okay? From this 1 and n is equal to 0, we have r times z of 0, right? And the times when n is equal to 0, you get P 0 + from this 1. When n is equal to 0, you get q0, when n is equal to 0, you get z0, okay? That is equal to, okay, r times r- 1 + p0 of r + q0 time z0, and that is equal to 0. Since we may the first coefficient z0 is not equal to 0, okay? I'm not explaining why, but I'd like to ask you to think about it, okay? Why is it there? Why we may assume that the very first coefficient to z0, is not equal to z0, okay? It's so easy the reasoning, okay? If z0 is not equal to 0 then for this equation to be equal to 0, you must have r times of r- 1 times p0 of r + q0. That must be a quarter 0, okay? It's a quadratic equation for the unknown r y, okay? I think that you may remind or something, okay? We give a name for the corresponding to the r of differential equation, okay? This is a kind of initial equation, right? Again, recorded as an individual equation. Of the given second order differential equation, and of which solution will be called? It's called the right? Because this is a quadratic equation in r, right? There might be here several different cases, right? Two distinct real There might be only 1 double Okay? There might be two complex conjugated roots, right? Anyway, so we call it as the initial equation, okay? And these solutions are called- Solutions are called Okay? Or sometimes they called it as exponents, okay? What is the p0 and q0? P0 is the very first coefficient tub, taylor series expansion of this one, okay? So here, what is the p0? This is x- x0 times p of x computing the x0, right? Or you'd better to write it as a limited when x goes to 0, x- x0 times p of x, right? Because when x is equal to x0, the value of this form might be an indeterminate form, like 0 over 0, it is possible, okay? What is the q0 by the same token? This is a limit of when x tends to x0, x- x0 times, what is it? Sq times q of x, right? The very first coefficient of Expanded of that function, okay? So whenever you have such second order differential equation with the given point is a regular singular point. You can get the initial equation of that, the differential equation directly by looking at the differential equation, okay? If you memorize, if you remember this two formulas it, okay? You don't have to plug in this summation of the the pm, x- x0 to the n, and the summation of z of n, x- x0 to the n + r, okay? You can immediately get this equation, and if you can obtain corresponding in okay?
