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| 科目：工 程 數 學 | （共1頁第1頁） |
| 1. | For a wave equation Initial condition  |
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| (a) | Please derive the d'Alembert's solution. | (15%) | ||
| (b) | If ,
show your solution. |
(5%) | ||
| 2. | Solve the following Legendre's equation (15%) |
(15%) | ||
| 3. | Integrate 
by the following processes |
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| (a) | Transfer the integral 
into 
by setting 
and get 
(10%) |
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| (b) | Solve 
by using the complex integral method. |
(15%) | ||
| 4. | Find the line integral 
along the given simple closed curve C=C1+C2+C3 as shown in the figure. Where
 ,
and C1:x2+y2=1;
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| C2: z2+y2=1; C3: x2+z2=1 | (10%) | |||
 |
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| 5. | System of differential equations as follows: Let  |
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| Find | (a) | the rank of A | (5%) | |
| (b) | the eigenvalues and eigenvectors of A | (10%) | ||
| (c) | the general solution of X | (5%) | ||
| 6. | 
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| (a) | Find the Fourier series of f(x) | (5%) | ||
| (b) | From the result of (a) to prove that | |||
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(5%) | |||
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