Re: [理工] 線代-微分方程
※ 引述《thank1984 (thankakimo)》之銘言：
: 題目： Let A = [0 -4], y(t) = [y_1(t)] , y''(t) = [y_1''(t)] . solve the
: [1 5] [y_2(t)] [y_2''(t)]
: system y''(t) = Ay(t) by the matrix method.
: Sol: A 之 eigenvalue : 1 , 4
: .
: .
: .
: .
: => X_1'' = X_1 => { X_1(t) = c1e^t + c2e^-t //***
: => X_2'' = 4X_2 { X_2(t) = c1e^2t + c2e^-2t //***
: 問題： 我想請問各位大大 ***的運算式做二次微分怎麼計算呢？？ 煩請解答 感謝
-1
先把A對角化 A =SDS D = | 1 0 | S = | -4 1 |
| 0 4 | | 1 -1 |
-1
令y=sx 代入 y"= Ay => sx" =sDs sx => x" = Dx
| x1"| = | 1 0 |
| x2"| = | 0 4 |
(D+1)(D+1)x1 = 0 => X_1(t) = c1e^t + c2e^-t
(D+2)(D-2)x2 = 0 => X_2(t) = c1e^2t + c2e^-2t
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