110年 - 110 國立臺灣大學_碩士班招生考試_ 工程科學及海洋工程學研究所丙組:工程數學(L)#100704

科目：研究所、轉學考(插大)◆工程數學 | 年份：110年 | 選擇題數：0 | 申論題數：8

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所屬科目：研究所、轉學考(插大)◆工程數學

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申論題 (8)

1. (25%) Show that the equation is exact, and obtain its general solution. Also, find the particular solution corresponding to the given initial condition as well.
  61270fa47335c.jpg
(a) x'=y,y'=-x
(b) x'=xy, y'=-x
(a) (1,3,2,0),(4,1,-2,-2),(0,2,0,3),(4,7,1,2)
(b) (2,0,1.-1,0),(1,2.0.3.D),(4,-4.3.-9,-2)

4. (20%) It is known that the nxn tridiagonal matrix

61270fea68032.jpghas eigenvalues 6127101c4fe06.jpg
for j=1,2,... ..,n. (A is called tridiagonal because all elements are zero except for those on the main diagonal and the two adjacent diagonals.)
Verify Bquation (1) by calculating the eigenvalues for n = 1 and n = 2.
(a) Is it possible for a matrix to have no eigenvalues? Explain.
(b) A given eigenvalue can have more than one LI eigenvector. Can a given eigenvector correspond to more than one eigenvalue? Explain.