[MUSIC] Let's remind you following, what we have done before, right? We were considering the nonhomogeneous problem, and to find the particular solution Xp, okay? We write the complementary solution X sub c as fundamental matrix times arbitrary constant vector, okay? And from this, we make a guess for the particular solution Xp as a fundamental matrix times the unknown column vector, capital u, right? Plugging this form into the differential equation, okay? So this Xp is a solution, okay? Capital P times unknown column vector u is a solution to this original problem. Capital P times the capital U prime is equal to capital F, right? And that's the equivalent to, okay, u prime, right? Capital U prime is equal to, okay, capital P inverse times the capital F, right, okay? Now we are ready to conclude it, right, okay? How to solve this is simple, forced to the system of equations, right, okay? That's trivial. So that means, okay, we must take capital U of t to be, right, to be the antiderivative of the inverse of capital V times capital F and dt, okay? Because this is an indefinite integral, okay, after finding this antiderivative, you must include the arbitrary constant, right? But, okay, we are looking for just any particular solution of this type. So, okay, we can take those arbitrary constant to be zero, right, for convenience, okay? I will explain it again through the concrete examples, right? Once we found this antiderivative of inverse of capital V times the capital F, right, we are ready to write down the particular solution Xp. What is Xp then? It's simple, right, okay? Capital V times capital U. Now capital U is given by inverse of capital V times capital F and dt, right? That's a particular solution. Finally, general solution of our original problem should be, right, okay? General solution will be X is equal to, okay, capital V times arbitrary vector C, okay, the arbitrary constant vector, okay, plus xp. That is equal to capital V times fundamental matrix times inverse of capital V times capital F and dt, okay? That's the general solution, okay? That's the general solution, okay? It's quite straightforward, right, okay? Through this approach, we can notice that the following, okay? Compare the two methods we have introduced so far, okay? And first the method of undetermined coefficients, and now the method of variation of parameters, right, okay, operational parameters. I think the advantage of the method of variation of parameters over the method of undetermined coefficients is rather the straightforward. Because in the case of method of undetermined coefficients, okay? We have two ways of finding the Xp, right, for the problem. The X prime is equal to AX + F, right, okay? And this method, for the case of undetermined, the coefficients, right, this nonhomogeneous term must be of the special form, right, okay? F must be, okay? There is a restriction, right, let me say just this way, right? There is some restrictions, right, restrictions on allowable Function, capital of K, right? To be precise, Ft must be a polynomial, or the exponential function, or the sines and cosines and their finite sums, or their finite products, right? For the operational parameters, no restriction, right? On the top of the Ft, right, okay? It can be applied to any nonhomogeneous term capital F, right? That's the immediate advantage, right? Even though in the case of the variation of parameters, we need to compute kind of the the indefinite integral, right? It might be a little bit tedious, but anyway, we have a strong advantage, right? The variation of parameters has a strong advantage over the method of undetermined coefficients, okay? Let's discuss some simple example, okay? As a first example, consider the following, the nonhomogeneous problem. X prime is equal to two by two system, 0- 1, 1, 0, and the X +, okay? And second of T and 0, right, okay? This is that, okay? First, you'd better to note the following, okay? In this nonhomogeneous system of equations, the method of undetermined coefficients is not good, okay? There is no way to make a reasonable guess for the particular solution using the indeterminant method of undetermined coefficients, right? Method of, Undetermined coefficients, Is not applicable, okay? Right, it's a good exercise for you to think about why, okay, why we cannot apply the method of undetermined coefficients to this problem, okay? Simply, this is not for the types of the functions for the nonhomogeneous, nonhomogeneous tone is allowed, okay? So we'd like to solve it. We'd like to find this particular solution by the method of evaluation of parameters, okay? First we must find the the excess of C, okay? So, the matrix A, okay, here the coefficient matrix 0, -1, and 1, 0, it has two Eigen values, okay? Two Eigen values, +-i, with corresponding Eigen vectors. Okay, with corresponding Eigen vectors 0, 1, and + or- I times up +1, 0, okay? That means what? You can write the complementary solution, okay? So that the corresponding homogeneous problem has two linear independent solutions, okay? Has two linear independent solutions. One is minus sine of T and the cosine of T, okay? And cosine of T and the sine of T, okay, they are, okay, two linearly independent solutions, okay? Solutions for what? For X prime is equal to 8 times X, right, corresponding homogeneous problem, okay? In other words, the fundamental metrics, okay? Now, the worst fundamental metrics is, okay? So therefore, fundamental metrics V of T is equal to two by two matrix given by the -sin t and the cosine of t, and the cosine of t and the sine of t, right, okay? This is the fundamental metrics, right? I claimed that the fundamental matrix is nonsingular everywhere, right, always, okay? Can you conform it, okay? What's the determinant of capital V? Product of the main diagonals. That is a -sin squared t minus the product of this subdiagonal. That is a- cosine cubed t, that is equal to -1, right? By the trigonometric identity, so that is never 0, right? We can confirm it, okay? This fundamental metrics is never singular, right? It's nonsingular everywhere, okay? Then our general theory said, okay, Okay, there is a particular solution Xp, okay, which is of, let me see, Which is of the time fundamental metrics V times antiderivative of inverse capital V times, this is the nonhomogeneous term, capital Ft and dt, right, okay? First, let's find the the inverse of capital V, okay? Can you find the inverse of this capital V, right. Okay, let's make the following comment, right, okay? If we have any nonsingular two by two matrix, right, finding inverse is very easy, right, okay? If we have a, b, c, d, nonsingular two by two matrix, its inverse is equal to, okay, one of the determinant. Determinant is ad- bc, okay? Then what, okay? Exchange this two, okay, that is a d and a. Then, okay, change the sign for the subdiagnol, -b and -c, right? That is the inverse, right, okay? As long as ad- bc, in other words, determinant is never 0, right, okay? That's the inverse here, right? It's so simple in the case of the two by two matrix, okay? So in the case of this Vt capital V, right, what's the determinant? 1 over the determinant is V. Already computed it, 1/ -1, right? Then, it's inverse of b. Okay, switch these two, sin t and- sin t, okay? Change the sign of these two parties, -cos t,- cos t, right? That's the inverse, right, okay? So this is equal to, this is -1. So that this is a- sin t, cos t, and the sine of t, right? In fact, these two by two matrix is the same as the original capital V, right, okay? So in our first example, okay, this Vt inverse is, inverse of the capital V is, again, capital V, okay? That's our conclusion way, right? So we now have, this is equal to capital V times, capital V inverse. That is, again, capital V. So that is the- sin t, cos t, cos t, sine t, times capital F. Capital F is, this is, the second t is 1/cos to, and 0, and dt, right, okay? The product is these two, compute, okay? Then this will be, okay? We have the same capital Vt. And the product of this will be, okay, minus sin t over cos t, Okay, times the 0. This is 1, okay, and 1, right? And dt, right, okay? We'll get this, okay? Is to find the antiderivative of this two functions, right, okay? And finally, making another product with the capital V. Then I'll skip the couple of steps here. Then this will be equal to- sin t log absolute value of cos t, and + t of cosine of t, okay? And cos t, log of absolute value of cos t + t sin t, right? That's a particular solution, okay? That's a particular solution of this problem. Then what is the, the general solution of this original problem will be, right? It's straightforward to conclude, right? Now the general solution X is equal to, okay? We already have this V, so let me use it this way, right? Capital V times, arbitrary constant vector +, okay, then this Xp, right, okay? Xp is down there, okay? This is a general solution, okay? Okay? [MUSIC]