Setting =x,x2=X,x3=x",…,xn=x-, (12） we have x=x2, (13) 为=X3, Xn-1=Xn) xn=F(t,1,x2,…,xn): 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
The initial conditions for(10) x(o)=k1,d(o)=k2,…,xa-)(o)=kn correspond to the initial conditions x(to)=k1,x2(to)=k2,…,xn(t0）=kn (14) for (12). 4口14①yt至2000 Linear of Differential Equations
The initial conditions for (10) x(t0) = k1, x 0 (t0) = k2,··· , x (n−1) (t0) = kn correspond to the initial conditions x1(t0) = k1, x2(t0) = k2,··· , xn(t0) = kn (14) for (12). Linear Systems of Differential Equations
If x is a solution of the initial value problem x四=Ft,x,,…,xa-1), x(to)=k1 (to)=k2;...x(n-1)(to)=kn 4日0,1无2000 Linear of Differential Equations
If x is a solution of the initial value problem x (n) = F(t, x, x 0 ,··· , x (n−1) ), x(t0) = k1, x 0 (t0) = k2,··· , x (n−1) (t0) = kn then (x1 = x, x2 = x 0 , x3 = x 00 ,··· , xn = x (n−1) ) is a solution the initial value problem x 0 1 = x2, x 0 2 = x3, ······ x 0 n−1 = xn, x 0 n = F(t, x1, x2,··· , xn), x1(t0) = k1, x2(t0) = k2,··· , xn(t0) = kn Linear Systems of Differential Equations
If x is a solution of the initial value problem x四=Ft,x,,…,xa-1), x(to)=kI,(to)=k2,...,x(-1)(to)=kn then (xI=x3,(1))is a solution the initial value problem x=x2, 名=x3, Xn-1=Xn: xm=F(t,x1,x2,…,xn), x1(to)=k1;x2(to)=k2,...,xn(to)=kn 4口4①，4元卡000 Linear of Differential Equations
If x is a solution of the initial value problem x (n) = F(t, x, x 0 ,··· , x (n−1) ), x(t0) = k1, x 0 (t0) = k2,··· , x (n−1) (t0) = kn then (x1 = x, x2 = x 0 , x3 = x 00 ,··· , xn = x (n−1) ) is a solution the initial value problem x 0 1 = x2, x 0 2 = x3, ······ x 0 n−1 = xn, x 0 n = F(t, x1, x2,··· , xn), x1(t0) = k1, x2(t0) = k2,··· , xn(t0) = kn Linear Systems of Differential Equations
On the other hand,if (x1,x2,3,..,n)is a solution of the initial value problem x=2, x=3, Xn-1 =Xn xn=F(t,x1,2,…,xn, x(to)=k1,x2(t0)=k2,…,xn(t0)=kn 4口4①，1无2)只0 Linear of Differential Equations
On the other hand, if (x1, x2, x3,··· , xn) is a solution of the initial value problem x 0 1 = x2, x 0 2 = x3, ······ x 0 n−1 = xn, x 0 n = F(t, x1, x2,··· , xn), x1(t0) = k1, x2(t0) = k2,··· , xn(t0) = kn then x1 is a solution of the initial value problem x (n) = F(t, x, x 0 ,··· , x (n−1) ), x(t0) = k1, x 0 (t0) = k2,··· , x (n−1) (t0) = kn Linear Systems of Differential Equations