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）︰2010年12月17日
考試時限（分鐘）：140分鐘
是否需發放獎勵金：是
試題 :
1.Basic concepts:(22%)
A.Define:
(1) Symmetric matrix; (2) Skew symmetric matrix; (3)Orthogonal matrix;
(4) Rank of a matrix
B.Using the following system as an example, answer (1)~(5)
CX=b (C,X,b皆為矩陣)
c11 c12 c13 b1
C=[c21 c22 c23] , b=[b2]
c31 c32 c33 b3
(1) Coefficient matrix; (2) Augmented matrix; (3)c23's cofactor C23;
(4)Describe Cramer's rule; (5) Describe Laplace expansion of det[A]
C.For the given eigenvalue problem:
AX=λX (A,X為矩陣)
a11 a12 a13 x1
A=[a21 a22 a23] , X=[x2]
a31 a32 a33 x3
explain:(1) What is eigenvector? (b)Why we have to start with
det(A-λX)=|A-λX|=0?
2.You have learned 3 different methods to solve for particular solutions of ODE:
method of undetermined coefficients; method of variation of parameters;
method of inverse operators. For each ODE, please determine the homogeneous
solution first and then use two methods to solve for the particular solution.
You will have to write down the general solution for each ODE.(21%)
A.(D^2-2D+1)y=x^2+(x^-2)e^x
B.y"-6y'+9y=(e^3x)/x
C.y"-y=(x^2)sin3x
3.Please find the general solutions of the following ODEs:(24%)
A.(x*D^4+D^3)y=100x^4
B.(x^3*D^3-x^2*D^2-7xD+16)y=9xlnx
C.(3x+2)^2*y"+3(3x+2)*y'-36y=3x^2+4x+2
4.Please fine the gerneral solution of the following system of ODEs:(8%)
x'+x+y'-y=e^t
x"+x'+x+y"-y'+y=t^2
(x'=dx/dt y'=dy/dt)
5.For the following eigenvalue problem:(13%)
AX=λX
a11 2 a13 x1
A=[a21 a22 5 ] , aij為實數 , X=[x2]
0 0 3 x3
trace A=6; det(A)=|A|=-30
A.Find the eigenvalues.
B.If a13=3 and a21=9, find a11, a12, and eigenvectors.
C.Please show that all eigenvectors form a linearly independent set.
6.A.What are the two methods that can be used to dertimine the inverse of a
matrix, please describe them?
B.Determine A^-1 (the inverse matrix of A).
-1 1 2
A=[ 3 -1 1]
-1 3 4
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