6. (8%) Solve
$[y^2 - e^x \sin(e^x + 1)]dx + 2xydy = 0.$

7. (8%) Solve $(x + 2y + 3)dx + (x - 2y + 4)dy = 0.$

8. (8%) Find the orthogonal trajectories of $(x - c)^2 + y^2 = c^2$.

9. (8%) Solve $\frac{d^2y}{dx^2} + y = x + \sec x.$

10. (8%) Find the Fourier series of the period function $f(x) = x + \pi, |x| < \pi$.

11. (10%) Solve the following initial value problem by the Laplace transform. $\begin{cases} y_1'(x) = y_1(x) - 2y_2(x) + 1, \\ y_2''(x) = -3y_1(x) - 2y_2(x), \\ y_1(0) = y_2(0) = y_2'(0) = 0. \end{cases}$

1. (20%) Completing the following traffic flow by using matrix multiplication. Calculate the number of cars leaving the network$$\vec{w} = \begin{bmatrix} w1 \\ w2 \\ w3 \\ w4 \end{bmatrix} \text{ given the number of cars entering the network } \vec{x} = \begin{bmatrix} 1000 \\ 2000 \end{bmatrix}.$$

2. (20%) Let A be an n × n matrix. Use elementary row operations to transform [A l_n] to [R B], where R is a matrix in reduced row echelon form. If R = ln, in which case A is invertible, show that B = A⁻¹.(Hint: Using the fact that there exists an invertible matrix P such that PA=R)

3. (40%) Prove that: An n × n matrix A is diagonalizable if and only if there is a basis for Rⁿ consisting of eigenvectors of A. Specifically, A = PDP⁻¹ if and only if the columns of P are the basis of Rⁿ and the diagonal entries of D are eigenvalues corresponding to the respective columns of P.

4. (20%) Let A be any \( m \times n \) matrix. Prove that \( A^T A \) and A have the same null space. (Hint: Let \( \vec{v} \) be a vector in \( \mathbb{R}^n \) such that \( A^T A \vec{v} = 0 \). Observe that \( A^T A \vec{v} \cdot \vec{v} = A \vec{v} \cdot A \vec{v} = 0 \))

1. (2% for each blank, 12% in total, 本題計算過程不計分) 　Suppose two infinite planes of charge are parallel to each other (Fig. 1), one positively charged and the other negatively charged. Both planes have the same charge density σ. The direction and the magnitude of electric field at point P1 (to the left of the positively charged plane) are      (1a)      and      (1b)     , respectively. The direction and the magnitude of electric field at point P3 (to the right of the negatively charged plane) are      (1c)      and      (1d)     , respectively. The direction and the magnitude of electric field at point P2 (between the two planes) are      (1e)      and      (1f)     , respectively.

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