推 asdf322505:請問板大 這題可以用微分運算符號ㄇ? 02/28 14:13
推 passguestjay:這種題目解法很多 微分運算符也可以解 書上應該有教 02/28 15:58
→ ntust661:你確定= =y 02/28 17:06
推 xxxorc:用運算子要經過很激烈的髂拉瑪吧? 02/28 19:18
※ 引述《asdf322505 ()》之銘言：
: y1'= -9y1+4y2-4y3
: y2'= -15y1+7y2-5y3
: y3'= 15y1-6y2+8y3
: ans; y1=2c1e^t+c3e^3t
: y2=5c1e^t+c2e^2t
: y3=c2e^2t-3c3e^3t
: 這題怎嚜解?
Y' = A Y
mt
Y = X e
mt
Y'= X m e
mt mt
X m e = A X e
m X = A X (eigenvalue problem)
-9-m 4 -4
│A - m│= [-15 7-m -5 ]
15 -6 8-m
3 3 2
= (-1) ( m - 6 m + ([-63+60]+[-72+60]+[56-30]) m - ([-9*7*8-20*15
-60*6 ]-[-60*7-60*8-30*9]) )
detA = (-504 - 300 - 360) + (420 + 480 + 270)
= -1164 + 1170 = 6
3 3 2
故特徵方程式 (-1) (m - 6m + 11m - 6 ) = 0
m = 1 , m = 2 , m = 3
是根!
特徵向量呢
-10 4 -4
│A - 1│= [-15 6 -5 ] , Rank(A - I) = 2
15 -6 7
-15 x1 + 6 x2 - 7 x3 = 0
-15 x1 + 6 x2 - 5 x3 = 0
x1 = 6 , x2 = 15 , x3 = 0
6 2
特徵向量 X = [ 15 ] = [ 5 ]
0 0
-11 4 -4
│A - 2│= [-15 5 -5 ] , Rank(A - 2I) = 2
15 -6 6
-15 x1 + 5 x2 - 5 x3 = 0
-15 x1 + 6 x2 - 6 x3 = 0
x1 = 0 , x2 = 1 , x3 = 1
0
特徵向量 X = [ 1 ]
1
-12 4 -4
│A - 3│= [-15 4 -5 ] , Rank(A - 3I) = 2
15 -6 5
-15 x1 + 4 x2 - 5 x3 = 0
-15 x1 + 6 x2 - 5 x3 = 0
x2 = 0 , x1 = -1 , x3 = 3
-1
特徵向量 X = [ 0 ]
3
mt
故解 Y = X e
2 t 0 2t -1 3t
Y = [ 5 ] c1 e + [ 1 ] c2 e + [ 0 ] c3 e
0 1 3
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