Here we have consider the problem of finding a solution about this, the ordinary point, we can extract the two linearly independent solution of the power series, the formulae. Now, with a lighter consider the problem of solving the linear differential equation around about singular point. As a motivation, let's consider the simplest possible such differential equations, say the Euler differential equation, say a ax squared y double prime plus a bxy prime plus cy is equal to 0. Where a is non-zero constant, and b and c are constant, but I'm assuming that b squared plus c squared is not equal to 0. The reason I'm assuming that the b squared plus c squared is not equal to 0 is rather obvious. Because it means the what? Either b or c is not equal to 0. If you write it into the standard form, then you will get y double prime plus a/b times b/a times 1/x, c/a times 1/x squared y and that is equal to the best is the standard form. Because I'm assuming a non-zero. Either b or c is non-zero so either this part is not identically equal to zero or that one is not identically equal to zero. That means what? Point x is equal to 0 is really a singular point of that Euler differential equation. The solution of the differential equation will be obtained by solving so-called the corresponding indicial equation. Corresponding indicial equation, say a times r times r minus 1, plus br plus c is equal to 0. I'm assuming that this indicial equation has an indicial root r which is real but not a non-negative integer. For this really indicial root r. We know that y is equal x squared is a solution to this Euler differential equation. Because I'm assuming that r is not a non-negative integer, this is a real but not all non-negative integer. Solution but which is not analytic at x is equal to 0. For example, would think about r is equal to 1/2, then x^1/2. This is not analytic at point x is equal to 0. For example, if we assume that r is equal to negative 1 integer but negative, then y is equal x negative 1. This is not analytic at point x is equal to 0. If r is not a non-negative integer, then y is equal x^r is a solution but which is not analytic at x is equal to 0. What does that mean? The function which is not analytic at x equal to 0, you cannot represent it as a power series. This is impossible by the definition of the analytic still. What does that mean to us? That means if we wanted to solve this whole differential equation around x is equal to 0. If you blindly try to find a solution of zero to infinity of C_nx^n. The solution of the power series a type. If we try a solution of that differential equation of this form, you definitely fail to recover the solution. Recovering this solution using this form is impossible. That means what? It fades away. What I mean is trying this power series formed either never work. We must think about some other ways of finding a solution of this differential equation around a singular point. That's the good motivation. To go further we need one equation relating to the singularity of the differential equation. Let me introduce the following definition. Let's go back to the original differential equation a_2(x) y double prime plus a_1(x) y prime plus a_0(x) y is equal to 0. Go back to that differential equation. Think about the singular point of this one or singular point x_0 of that differential equation. With a lighter distinguish to introduce two different types of these singular points. This is the regular singular point. If x minus x_0 times the little p(x) and x minus x_0 square times q(x) both are analytic at point x_0. What I mean by little p(x) and the q(x) that's the notation I used in introducing the standard form of the differential equation. In other words, this is a_1 over a_2, q(x) is equal to a_0(x) the a_2(x). If you multiply this little p(x) by x minus x_0 and if multiply this little q(x) by x minus x_0 square and if both these two functions are analytic at point x_0 then I call the singular point x_0 to be a regular singular point. Singular, bad but not that much bad, not really bad point. Otherwise, call x_0 to be our irregular singular point. I'm introducing these two notions. Singular point is the bad point in some sense bad, my really bad regular singular point. Truly bad one irregular singular point. Yeah, as a concrete example, let's consider the following. Second-order differential equation with polynomial coefficients x squared minus 4, and square y double prime, minus 2x plus 2y prime, plus 3y is equal to 0. What I'd like to do is classify all real values as an ordinary point or the regular singular point or irregular singular point. [inaudible] it as the standard form, then you get y double-prime minus 2 over x plus 2. X squared minus 4 squared is equal to x minus 2 squared, x plus 2 squared, and the y prime plus 3 over x minus 2 squared and x plus 2 squared, and the y is equal to 0. First, let's see where is the one common factor y, this x plus 2, and the 1x plus, they canceled out. We get there finally. In its standard form, px is equal to minus 2 over x minus 2 squared, and x plus 2, and the q of x is equal to 3 over x minus 2 squared and x plus 2 squared. We can immediately see that. The singular points of the differential equation are, x is equal to 2 or minus 2. We have a data is a two singular points. X is equal 2 or x is equal to minus 2. Or others are ordinarily points. For this two singular points, let's determine whether they are regular singular points or irregular singular points. First, let's think about the axis equal to minus 2 for this singular point. As it the regular singular point or irregular singular point, it depends on the analyticity of the following functions. So x plus 2p of x. What is it? There is a minus 2 over, x minus 2 squared. This is our tribute analytic. Had x is equal 2 minus 2. On the other hand, x plus 2 square q of x is equal to 3 over x minus 2 squared. This is also analytic at x is equal to minus 2. Therefore we conclude saw on analyticity minus two, this is a regular singular point. Now let's consider the x is equal to plus 2 , another singular point. To see that, whether this is a regular or irregular singular point. Then it depends on the analyticity of x minus 2p of x. First, this is equal to 1. X minus 2 times the p of x is equal to minus 2 over x minus 2, x plus 2. Is it an analytic x is equal to distributed not way. It's not analytic, x is equal to 2. We do not need to look at the x minus 2 squared times q of x. Already this factor, then means what? Point x is equal 2 is an irregular singular point. I hope by this example, you may have the better idea, how to distinguish the regular singular point and irregular singular point.