On the other hand,if (1,x2,x3,..,n)is a solution of the initial value problem x=2, x=3, Xn-1=Xn; xmn=F(t,x1,2,…,xn), x(to)=k1,x2(t0)=k2,…,xn(i0)=kn then x is a solution of the initial value problem x四=Ft,x,X,…,xn-1, x(o)=k1,(o)=k2,…,xa-(o)=kn 4口4①，4元卡000 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
Example d3x 3w- dx dx = x(to） =k1,X(to)=k2,X"(to)=3 4口14①yt至2000 Linear of Differential Equations
Example d 3 x dt3 = 3tx− dx dt + d 2 x dt2 x(t0) = k1, x 0 (t0) = k2, x 00(t0) = k3 Setting x1 = x, x2 = x 0 , x3 = x 00 , we have x 0 1 = x2, x 0 2 = x3, x 0 3 = 3tx1 −x2 +x3. x1(t0) = k1, x2(t0) = k2, x3(t0) = k3 Linear Systems of Differential Equations
Example dx dx dx 3- x(to) =kI,(to)=k2,x"(to)=k3 Setting 灯=x,X2=x,x3=x", 4口14①y至元2000 Linear of Differential Equations
Example d 3 x dt3 = 3tx− dx dt + d 2 x dt2 x(t0) = k1, x 0 (t0) = k2, x 00(t0) = k3 Setting x1 = x, x2 = x 0 , x3 = x 00 , we have x 0 1 = x2, x 0 2 = x3, x 0 3 = 3tx1 −x2 +x3. x1(t0) = k1, x2(t0) = k2, x3(t0) = k3 Linear Systems of Differential Equations
Example dx dx d 3x- x(to) = kI (to)=k2;x"(to)=k3 Setting 灯=x,x2=x,3=X", we have 好 =X2, 为 三 X3, 名 =31-x2+x3 4口14①y至元2000 Linear of Differential Equations
Example d 3 x dt3 = 3tx− dx dt + d 2 x dt2 x(t0) = k1, x 0 (t0) = k2, x 00(t0) = k3 Setting x1 = x, x2 = x 0 , x3 = x 00 , we have x 0 1 = x2, x 0 2 = x3, x 0 3 = 3tx1 −x2 +x3. x1(t0) = k1, x2(t0) = k2, x3(t0) = k3 Linear Systems of Differential Equations
Example 3x- dx dx d而 = x(to) = kI (to)=k2;x"(to)=k3 Setting 灯=x,2=x,x3=X", we have x =X2, 名 3, 名 3x1-x2+X3 x1(to) =k1,x2(to）=k2,x3(to）=k3 4口14①y至元2000 Linear of Differential Equations
Example d 3 x dt3 = 3tx− dx dt + d 2 x dt2 x(t0) = k1, x 0 (t0) = k2, x 00(t0) = k3 Setting x1 = x, x2 = x 0 , x3 = x 00 , we have x 0 1 = x2, x 0 2 = x3, x 0 3 = 3tx1 −x2 +x3. x1(t0) = k1, x2(t0) = k2, x3(t0) = k3 Linear Systems of Differential Equations