45.Method of variation of constants

The general form of system of differential equations: (1)

Let`s introduce a linear operator L(y)= , then (1) we can write in the following form: L[y]=F (2). If F=0 , then the operator equation (2) is called homogeneous and has the form: L[y]=0 (3)

Let Y= - the general solution of homogeneous sytem (3). Let`s find a solution of (2) in following way: Y(x)= , where Ci(x - unknown functions. Substituting the solution into (2):

, and tak into a consideration that Yi - solutions of (3), it means . We get, - vector equation. The last correspondence can be written in the form of n-equations with n-indeterminates Ci(x). Meanwhile W≠0 on [a,b], because Yi - fundamental system of solutions of (3) and, hence, we can explicitly determine unknown functions: Ci’(x)= i= , Ci’(x)= And then the general solution of (2) has a form:Y(x)= .