107年 - 107 國立嘉義大學_碩士班招生考試試題_生物機電工程學系：工程數學#146097

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 \( F = yi + 2xj \) 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 \[ \frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \dots \] (10%)