4一阶线性常系数微分方程组 dxi(t a1()+a12x2(D)+…+a1nxn(t)+f1(1) dt dx2(t dx=的2ux1(t)+a2(+…+a2nxn()+/2( dxn(t) anIx(t)+an2x2(t)+.+annen(t)+fn(t) dt 满足初始条件x(0)=C,i=1,2,…,n
4 一阶线性常系数微分方程组 1 11 1 12 2 1 1 2 21 1 22 2 2 2 1 1 2 2 d ( ) ( ) ( ) ( ) ( ) d d ( ) ( ) ( ) ( ) ( ) d d ( ) ( ) ( ) ( ) ( ) d n n n n n n n nn n n x t a x t a x t a x t f t t x t a x t a x t a x t f t t x t a x t a x t a x t f t t = + + + + = + + + + = + + + + 0 ( ) , 1,2, , 满足初始条件 x t c i n i i = =
dx(t =Ax(t)+∫(t) →dt x(to)=C 其中,A=Vxm,x()=(x(t),x2(),…,x( c=(a1,c2,…,cn)1,f()=(f(),2(t)…,fn() →(ex(1)=e(-4)x(+4dx(t) dr =e Atr delt d4x(t)=eAf)在,1上积分
0 d ( ) ( ) ( ) d ( ) x t Ax t f t t x t c = + = ( ) , 其中，A a = ij n n 1 2 ( ) ( ( ), ( ), , ( )) , T n x t x t x t x t = 1 2 ( , , , ) , T n c c c c = 1 2 ( ) ( ( ), ( ), , ( )) , T n f t f t f t f t = d d ( ) ( ( )) ( ) ( ) d d At At At x t e x t e A x t e t t − − − = − + d ( ) ( ( )) ( ) d At At x t e Ax t e f t t − − = − = 0 在 上积分 [ , ] t t
e-Alx(0-e Aox(t0)=eAt f(cjd →x0=+c+ur 例1:求解初值问题 dx(t) x1(t)-2x2(t)+6x3(t)- dt dx2(t) =-x1()+3x3() dt dx3() dt x1()-x2()+4x3()+e 1(0)=1,x2(0)=0,x3(0)=0
1 1 2 3 2 1 3 3 1 2 3 1 2 3 d ( ) ( ) 2 ( ) 6 ( ) d d ( ) ( ) 3 ( ) d d ( ) ( ) ( ) 4 ( ) d (0) 1, (0) 0, (0) 0 t t x t x t x t x t e t x t x t x t t x t x t x t x t e t x x x = − − + − = − + = − − + + = = = 例 1： 求解初值问题0 0 0 ( ) ( ) ( )d At t At A t e x t e x t e f − − − − = 0 0 ( ) ( ) ( )d A t t t At A t x t e c e e f − − = +
解 1-26 A=-103,c=0,f(r)=0|→ 1-2t-2t6t At =e 3t, t1+3t 1-2t 1-87 e c=e-t en f( dE 4τdz→ 0
解： 1 2 6 1 1 0 3 , 0 , ( ) 0 1 1 4 0 t t e A c f t e − − − = − = = − − 1 2 2 6 1 3 , 1 3 At t t t t e e t t t t t t − − = − − − − + 1 2 At t t e c e t t − = − − 0 ( )d t A e f − 0 1 8 4 d 1 4 t − − = − −
7 t-2t 2t2+2 x(t=e C+eat rt e f∫(z)dτ 1-3t+4t x(t=ec+ At e f(rdr=e-t+2r 2
0 ( )d t A e f − 2 2 2 4 2 2 t t t t t − − = − − 0 ( )d t At A e e f − 2 2 2 4 1 2 2 2 t t e t t − = + 0 0 ( ) ( ) ( )d A t t t At A t x t e c e e f − − = + 0 ( ) ( )d t At At A x t e c e e f − = + 2 2 2 1 3 4 2 2 t t t e t t t − + = − +