Except some first order differential equations we have studied in Chapter 2, we do not have many analytic methods of solving general high order non-linear differential equations. 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 our service to second order equation for simplicity. First, let's consider the linear differential equations. Consider linear nth order differential equation. Equation number 1, which is a_n of x times n derivative of y plus a_n minus 1 of x times n minus 1 derivative of y plus dot, dot, dot plus a_1 of x times y prime plus a_0 of x times y that is equal to b of x. Where the coefficients a_n of x, and a_n minus 1 of x, and a_0 of x, and b of x, they are all continuous functions on some interval I. When all the coefficients from a_n, and a_n minus 1 through a_0, if they are constants, we say that the differential equation one has constant coefficients. In that case, this right-hand side of b of x, it did not become the constant. We're only requiring that the coefficient of the unknown functions of y or any of these derivatives. All those coefficients say a_n of x through the a_0 of x, if they are all constants, then we say that equation is constant coefficients otherwise it has variable coefficients. When the right-hand side, say b of x, right here, if this is identically 0 on the interval I, then we say that the equation is homogeneous. Otherwise, if bx is not identically equal to 0 on I, then we say that differential equation is non-homogeneous. In this chapter, we always assume that a_n of x, which is called the leading coefficient of differential equation is never 0 on the interval I. Then we can divide the whole equation by a_n of x so that we can rewrite the differential equation 1 its so-called standard form. Say n derivative of y plus p_n minus 1 of x, n minus 1 derivative of y, p_1 of x, y prime, p_0 of x y, that is equal to g of x where the coefficients p_k of x is a_k of x over a_n of x, and g of x is bx over a_n of x. This is the so-called standard form in the sense that it's leading coefficients, the coefficients of highest derivative of y, which is equal to identically 1. Then we call it as a standard form of the differential equation. With the notation, capital D^k y is equal to kth derivative of y for any non-negative integer k. Then we may express the given differential equation 1 as L of y is equal to b of x. What is L? L is equal to a_n of x times D^n plus a_n minus 1 of x times D to the n minus 1, and a_1 of x times D, and plus a_0 of x. This is so-called the linear differential operator, or just the differential operator. Such differential operator is linear in the sense that, what is the differential operator 2? Here I will remind it to you. Differential operator L, this is equal to a_n of x and, D^n plus and so on, a_1 of x, and D plus a_ 0 of x. Recall that the capital D means simply d over dx. We say that this differential operator L is linear in the sense that L of the Alpha times f of x plus Beta times the g of x is equal to Alpha times L of f of x plus Beta times L of g of x for arbitrary two functions, f and g, and arbitrary two constants, Alpha, and Beta. This property you can check it very easily, for this differential operator. The property 3 is called the linearity of the differential operator. In that sense that we call the operator L a linear differential operator.
