作者youmehim (哩挖伊)
看板Grad-ProbAsk
标题Re: [理工] [工数]-拉普拉斯.....
时间Fri Sep 18 22:52:46 2009
3.
y" + 3y' + 2y = f(t) , y(0) = y'(0) = 0 , f(t) = 4t 0<t<1
8 1<t
f(t) = 4t[u(t) - u(t-1)] + 8 u(t-1) = [-4(t-1) + 4]u(t-1) + 4t u(t)
L{y} = U(s)
L{y'} = sU - y(0) = sU
2 2
L{y"} = s U - sy(0) - y'(0) = s U
-s -4 4 4
L{f(t)} = e [ ----- + --- ] + -----
2 s 2
s s
2 -s -4 4 4
( s + 3s + 2 )U = e [ ----- + --- ] + -----
2 s 2
s s
-s -4 4 4
U = e [ -------------- + ------------- ] + --------------
2 s(s+1)(s+2) 2
s (s+1)(s+2) s (s+1)(s+2)
-s 3 -2 -4 1 2 -4 2
= e [ ( --- + ----- + ----- + ----- ) + ( --- + ----- + ----- ) ]
s 2 s+1 s+2 s s+1 s+2
s
-3 2 4 -1
+ ( --- + ----- + ----- + ----- )
s 2 s+1 s+2
s
-1 -1 -s -t -2t -t -2t
L {U} = L { e [L( 3 - 2t - 4e + e + 2 - 4e + 2e )]}
-t -2t
+ ( -3 + 2t + 4e - e )
-1 -s -t -2t -t -2t
= L { e [L(5 - 2t - 8e + 3e )]} - 3 + 2t + 4e - e
-(t-1) -2(t-1) -t -2t
= [5 - 2(t-1) - 8e + 3e ]u(t-1) - 3 + 2t + 4e - e
-t+1 -2t+2 -t -2t
= [7 - 2t - 8e + 3e ]u(t-1) - 3 + 2t + 4e - e
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1F：推 mdpming:好详细的解答~~可以好好研究了!! 09/18 22:58
※ 编辑: youmehim 来自: 218.173.131.250 (09/18 23:08)