課程名稱︰工程數學一
課程性質︰系必修
課程教師︰王勝仕
開課學院：工學院
開課系所︰化工系
考試日期（年月日）︰2011年1月14日
考試時限（分鐘）：140分鐘
是否需發放獎勵金：是
試題 :
1.Find the inverse Laplace transform of F(s)=L{f(t)}
(1)F(s)=((e^-2πx)-(e^-8πx))/(s^2+1)
(2)F(s)=(s(s^2-9))^-1
(3)F(s)=(3s+2)/(4s^2+12s+9)
(4)F(s)=ln((s+a)/(s+b))
(5)F(s)=(s^2+4s+13)^-2
2.Find the Laplace transform of f(x)
(1)f(t)=(cosh(t/2))^2+sin(3t-1/2)
(2)f(t)=(e^at)*(sinht)/t
(3) f(t)
｜
｜
｜_ _
1 ｜ ／＼
｜／ ＼
______________(t)
1 2
(4) f(t)
｜
｜
｜_ _
1 ｜ ／＼ ／＼ ／＼ ....
｜／ ＼／ ＼／ ＼
______________________________(t)
(5)f(t)=(t^2+1)u(t-1)
3.(1)a)Define Dirac delta functions/unit impulse function δ(t-c) using unit
∞
step functions; b)Determine the value of ∫δ(t-c)dt (where 0<c<∞);
c)Find the Laplace transform of δ(t-c) (9%)
(2)Using two ways to find the Laplace transform of ((s+a)(s+b))^-1(6%)
4.Solve the given system of ODE using the method involving matrix analysis:(10%)
y1'=4y2+9t
y2'=-4y1+5
5.You are given the following matrix A:(15%)
2 2 1
A=[1 3 1]
1 2 2
(1)Find the eigenvalues and eigenvectors of AX=λX.
(2)If A is similar to D(a diagonal matrix), waht is the transition matrix P
and its inverse P^-1?
(3)Please show A^5
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