107年 - 107 國立嘉義大學_碩士班招生考試試題_土木與水資源工程學系：工程數學#146539

1. Find the solution of the equation \( y''-4y'+4y=0 \) for which \( y=3 \) and \( y'=4 \) when \( x=0 \). (25%)

2. Calculate the work done by the force \( F = y\mathbf{i} + 2x\mathbf{j} \) along the straight line from point (0, 0) to point (1, 1). (25%)

(a) Find the eigenvalues and eigenvectors of A. (10%)

(b) Find the eigenvalues and eigenvectors of \( A^{-1} \). (5%)

(c) Find an orthogonal matrix \( P \), that is \( P^{-1}=P^T \), such that \( P^{-1}AP=D \), where \( D \) is the diagonal matrix of the eigenvalues of \( A \). (10%)

(a) Find the Fourier series of on the given interval. (15%)

(b) Use the result to show the sum of the series \[ \frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots. (10\%) \]