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# 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

【非選題】
1.

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.

【非選題】
2.
2. (20%) Determine the equation of the phase trajectories for the given systen, and sketch several representative trajectories. Use arrows to indicate the direction of movement along those trajectories.

【題組】(a) x'=y,y'=-x

【非選題】
3.【題組】(b) x'=xy, y'=-x

【非選題】
4.
3. (25%) Determine whether the following set is LI (Linear independence) or LD (Linear Dependence). If it is LD, then give a linear relation among the vectors.

【題組】 (a) (1,3,2,0),(4,1,-2,-2),(0,2,0,3),(4,7,1,2)

【非選題】
5.【題組】(b) (2,0,1.-1,0),(1,2.0.3.D),(4,-4.3.-9,-2)

【非選題】
6.

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

has eigenvalues
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.

【非選題】
7.
5. (10%)

【題組】 (a) Is it possible for a matrix to have no eigenvalues? Explain.

【非選題】
8.【題組】(b) A given eigenvalue can have more than one LI eigenvector. Can a given eigenvector correspond to more than one eigenvalue? Explain.

### 懸賞詳解

#### 國三數學下第一次

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