作者sulanpa (...)
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标题Re: [微积] 请问一题一阶常微分方程组的解
时间Sat Dec 25 19:19:53 2010
※ 引述《shinehsnu (张小光)》之铭言：
: 大家好
: 我在算一题最佳化问题
: 需要解一个一次常微分方程组
: 题目如下
: (L1)' = 5*L2
: (L2)' = -L1 + 2*L2
: L1(1) = 0, L2(1) = 0
: 请问这样的题目要怎麽求解
: 或是MATLAB是否有指令可以直接计算出来
: 因为我不是数学系的
: 所以来请教一下
: 谢谢大家
(L1)' = 5L2 ... (1)
(L2)' = -L1 + 2L2 ... (2)
by(2)
(L2)'' = -(L1)' + 2(L2)' = -5L2 + 2(L2)'
=> (L2)'' - 2(L2)' + 5L2 = 0
characteristic equation
r^2 - 2r + 5 = 0
=> r = (2 ±sqrt[4-20])/2 = 1 ±2i
=> L2(t) = e^[t] (c1cos2t + c2sin2t)
(L2)'(t) = e^[t] (-2c1sin2t + 2c2cos2t) + e^[t] (c1cos2t + c2sin2t)
= e^[t] ([c1+2c2]cos2t + [-2c1+c2]sin2t)
substitute in (2)
=> L1 = e^[t] ([c1-2c2)cos2t + [2c1+c2]sin2t)
initial condition
L1(1) = 0 and L2(1) = 0
gives
0 = e([c1-2c2]cos2 + [2c1+c2]sin2)
0 = e(c1cos2 + c2sin2)
=> c1 = c2 = 0
=> L1(t) = 0 and L2(t) = 0
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