Let me introduce the following concepts, okay? Considered a differential equation for again, which is linear and nth order homogeneous differential equation. Say, A sub of x times and steady vertical y plus A7 minus 1 of X time and minus 1 derivative of y plus that it's one of X times Y plus a sub 0 of X times Y. That is equal to zero, okay? Linear and nth order the variable coefficient homogeneous differential equation. Well, I assume that. Okay? Okay, sure. All the coefficients AI of X, okay? They are continuous, okay? On some interval I and the leading coefficient, okay? Ace of N of X, right? Ace of N of X it's never zero on the interval I, okay? For this differential equation 4, okay? We call set of n and linearly independent solutions of the differential equation, okay? On an interval I, okay? We call it a fundamental set of solutions of differential equation for on the interval I, okay? You must know to that the order of this differential equation which is the end, okay? It must coincide the number of linearly independent solutions, okay? We should have exactly and linear independent solutions of that differential equation, okay? And we call the set y sub of I where I moving from one to the end to be a fundamental set of solutions of differential equation for on this interval, okay? And it's such a differential equation, okay? Always has a fundamental set of solutions as the following throw them shows. Existence of a fundamental set of solutions, okay? Any linear homogeneous differential equation 4, L Y is equal to zero, okay? I hope you remind what is that, okay? L of y. It's nth order linear differential equation, okay? Always as a fundamental set of solutions on I. It always have a fundamental solution. Improving this theory. Again, I'm assuming that n is equal too to make a thing that's simple. And by theorem 4.2, right? That's theorem 4.2. It's a unique existence of solution for initial value problem, right? So, by the theorem choose arbitrary point tax zero in the interval I, and to consider the following two initial value problems, okay? Differentially questions are the same thing. The second order linear homogeneous differential equation. First initial value problem requires the initial condition to be Y of X0 is equal to 1 and the y prime of X0 is equal to 0. Second initial value problem requires, y of 0 is equal to 0. Y Prime of x 0 is equals to 1, okay? Post this first and the second initial value problems 534.2. They should have a unique solution. Say, y x of 1 and x y of 2 on the interval I, right? That is guaranteed by the theorem 4.2, right? For these two solutions of Y1 and Y2. Y 1 is a solution to this Initial value problem A first, Y 2 is a solution to the second initial value problem. Computed here run skin at point X0, okay? That's the determinant of this 2 by 2 matrix is right. How much is this one? That is 1. How much is this one? Because this is a solution to the second problem, that is zero. What is the Y one prime of X zero. That is zero from this initial condition. What is the Y two prime of X zero? Then must be equal to run by this initial condition. So, our 2 by 2 matrix is 1 0, 0 1, right? It's determinant 1 which is not equal to 0 way. So, by the theorem 4.3 that we have just approved, right? These two solutions y of one and the y of 2 a linearly independent on I way and the solutions of differential equation, right? Because this is a second order, and we now however, two linearly independent solution, right? That we call a fundamental solution of the equation, okay? So we are done, okay? So, at least for little n is equal to, right? We confirm that there must be a fundamental set of solutions, right? For example. Okay. An example, one. Okay. We've handled it before. We have seen that y of one is equal to x, and the y of 2 is equal to x times the log of x. Both of them are solutions of this variable coefficient linear second order homogeneous differential equation on the interval from zero to infinity, right? Can you remind me? Right. That's a computer they run skin, right? Run skin of W of X 1 and the Y 1 and the Y 2, right? What is it? This is the Y 1, which is X, right? And y2 that is x times the log of x, right? And its derivative with 1. It's derivative will be x the log X plus 1, right? Okay. How much is the determinant? Right. X Log of X plus X minus X log of X that is X, right? This is an ever zero on the interval from zero to infinity, right? What does that mean? Right. By the run skin test, okay? The two solutions X and X times log of X is a fundamental set of solutions to given homogeneous second order differential equation on the interval from 0 to infinity, right? It can come for me.