3. Solve \( (2e^{2x+y} + 3ye^y)dx + (3xe^y - 4)dy = 0 \quad (10\%) \)

4. Expand the periodic function f(x) shown in Fig. 1 in a Fourier trigonometric series. (10%)

5. Solve \( y(t) - \int_{0}^{t} y(\tau)(t - \tau) d\tau = 2 - \frac{1}{2}t^{2} \) (10%)

6.Find the Fourier transform of \( f(x) = xe^{-x^2} \) (10%)

7. Solve \( y'' + 2y' - 24y = 8 - (x + 2)e^{4x} \) (10%)

8.The following system of equations has unique solutions. Solve this system using the method of Gauss-Jordan elimination with matrices. (10%) \[ \begin{aligned} x - 2y + 3z &= 9 \\ -x + 3y &= -4 \\ x - 5y + 5z &= 17 \end{aligned} \]

9. Use the matrix inverse method to solve the following system of equations. (10%)2x + 3 y + z = −13x + 3 y + z = 12x + 4y + z = −2

10. Find the determinant of A below (10%) \[ \mathbf{A}= \begin{bmatrix} 1 & -2 & 3 & 0 \\ -1 & 1 & 0 & 2 \\ 0 & 2 & 0 & 3 \\ 3 & 4 & 0 & -2 \end{bmatrix} \]

1. Translate the paragraphs into Chinese.A. Many freshwater resources are contaminated through human activities. The major types of water pollution include excessive nutrition produced from agriculture, toxic chemicals such as pesticides and metals, and thermal pollution produced from thermal power plant. (25%)

B. Circulatory systems of vertebrate are composed of blood, blood vessel, and the heart, and evolved as a means to transport oxygen and nutrients rapidly through the body. Blood are complex fluids consisting plasma and cells, which are involved in oxygen transport, immune responses and blood clotting. (25%)

C. Hybrid tilapia is a relatively inexpensive seafood source for common households in Taiwan. To maintain low production cost, the protein content of a typical local commercial tilapia diet is approximately 250 g kg-1. Furthermore, it is common practice to totally substitute the plant protein source, mainly soybean meal, for fish meal to reduce the cost of tilapia feed. Therefore, the anti-nutrients in soybean meal may influence the digestibility of the diet to tilapia. (25%)

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