Definition A solution of the system (8)is an ordered set of functions (x1,x2,..)that satisfies the system on some interval. Definition A set of auxiliary conditions of the form x1(to)=k1,x2(0)=k2,…,xn(t0)=kn, (9) where the ki are given numbers,is called a set of initial con- ditions for the system.The system together with the initial conditions is called an initial value problem. 4口14①yt至2000 Linear of Differential Equations
Definition A solution of the system (8) is an ordered set of functions (x1, x2,··· , xn) that satisfies the system on some interval. Definition A set of auxiliary conditions of the form x1(t0) = k1, x2(t0) = k2,··· , xn(t0) = kn, (9) where the ki are given numbers, is called a set of initial conditions for the system. The system together with the initial conditions is called an initial value problem. Linear Systems of Differential Equations
Transformation 1 How to rewrite the initial value problem of a single nth order differential equation for one unknown function, 4口4①，1无2)只0 Linear of Differential Equations
Transformation 1 How to rewrite the initial value problem of a single nth order differential equation for one unknown function, of the form x (n) = F(t, x, x 0 ,··· , x (n−1) ), (10) x(t0) = k1, x 0 (t0) = k2,··· , x (n−1) (t0) = kn (11) as the initial value problem of a system of first-order differential equations for n unknown functions of the form (8)? Linear Systems of Differential Equations
Transformation 1 How to rewrite the initial value problem of a single nth order differential equation for one unknown function,of the form x回=Fx,,…,xa-, (10) x(to)=kI,(to)=k2;...x(-1)(to)=kn (11) 4日0,1无2000 Linear of Differential Equations
Transformation 1 How to rewrite the initial value problem of a single nth order differential equation for one unknown function, of the form x (n) = F(t, x, x 0 ,··· , x (n−1) ), (10) x(t0) = k1, x 0 (t0) = k2,··· , x (n−1) (t0) = kn (11) as the initial value problem of a system of first-order differential equations for n unknown functions of the form (8)? Linear Systems of Differential Equations
Transformation 1 How to rewrite the initial value problem of a single nth order differential equation for one unknown function,of the form x回=F,x,X,…,xm-l), (10) x(to)=kI,(to)=k2,..,x(n-1)(to)=kn (11) as the initial value problem of a system of first-order dif- ferential equations for n unknown functions of the form (8)? 4口10,41元3000 Linear of Differential Equations
Transformation 1 How to rewrite the initial value problem of a single nth order differential equation for one unknown function, of the form x (n) = F(t, x, x 0 ,··· , x (n−1) ), (10) x(t0) = k1, x 0 (t0) = k2,··· , x (n−1) (t0) = kn (11) as the initial value problem of a system of first-order differential equations for n unknown functions of the form (8)? Linear Systems of Differential Equations
Setting 灯=x,x2=X,x3=x”,…,xn=xn-l), (12) 4口4①，1无2)只0 Linear of Differential Equations
Setting x1 = x, x2 = x 0 , x3 = x 00 ,··· , xn = x (n−1) , (12) we have x 0 1 = x2, (13) x 0 2 = x3, ······ x 0 n−1 = xn, x 0 n = F(t, x1, x2,··· , xn). Linear Systems of Differential Equations