[MUSIC] Okay, except some first order differential equations, we have studied in chapter two. We do not have many analytic methods of solving general high order non-linear differential equations. So here in Chapter 4, we introduce the general theory of linear high order differential equations including methods of solving constant coefficient equations. In most cases, we confine ourselves to second order equation for simplicity. So, first, let us consider the linear differential equations, okay? Consider linear nth order differential equation, equation number 1, okay? Which is a sub n(x) times n derivative of y + a sub n- 1(x) times n- 1 derivative of y + ..., + a sub 1(x) times y prime and + a sub 0(x) times y, that is = b(x), okay? And where the coefficients a sub n(x), and a sub n- 1(x), and a0(x), and b(x), they are all continuous functions on some interval I, okay? When all those, the coefficients, from a sub n and a sub n- 1 through a0, if they are constants, we say that the differential equation 1, okay? Has constant coefficients, okay? In that case, this right-hand side of b(x), right? It need not be constant, okay? We are only requiring that the coefficient of the unknown functions of y, or any of these derivatives, right? All those are coefficients, say a sub n(x) through the a0(x). If they are all constants, then we say that equation is constant coefficients, otherwise, it has a variable coefficients, okay? When the right-hand side say b(x) right here, if these is identically 0 on the interval I, then we say that the equation is homogeneous, okay? Otherwise, if b(x) is not identically equal to 0 on I, then we say that differential equation is nonhomogeneous, okay? In this chapter, we always assume that a sub n(x), which is called the leading coefficient of differential equation, is never 0 on the interval I, okay? Then we can divide the whole equation by a sub n(x), right? So that we can rely to the differential equation (1) in its so-called standard form, okay? Say n is derivative of y + p sub n- 1(x), n- 1 derivative of y, p sub 1(x) y prime, p sub 0(x) y that is = g(x), where the coefficients p sub k(x) is a sub k(x) over a sub n(x), right? And g(x) is b(x) over a sub n(x), right, okay? This is the so called the standard form in the sense that it's leading coefficients. The coefficients of highest derivative of y, okay, which is equal to identically 1, okay? Then we called as a standard form of the differential equation, okay? With the notation, capital D to the ky = k is derivative of y for any non-negative integer k, then we may express the given deferential equation (1) as, right, l(y) = b(x), right? What is then, L = a sub n(x) times D to the n + a sub n- 1(x) times D to the n-1, and a sub 1(x) times D + a sub 0(x), right? This is so called the linear differential operator, okay? Or just the differential operator, okay? Such a differential operator, okay, is linear in the sense that what is the differential operator 2? Here, I will remind it to you, okay? Differential operator to L = a sub n(x) and D to the n + and so on, a sub 1(x) and D and + a sub 0(x), right? We call that the capital D means simply d over dx, okay? So, we say that this differential point to L is a linear, in the sense that, okay, L of the alpha times f(x) + beta times g(x) = alpha times L[f(x)] + beta times L[g(x)] for arbitrary two functions f and g and arbitrary two constants alpha and beta, right? Okay? This property you can check it very easily, okay? For this differential operator, okay? And the property three is called the linearity of the differential operator okay? In that sense, we call the operator L, right? A linear differential operator, okay?
