108年 - 108 國立嘉義大學_碩士班(食品科技組)招生考試試題_機械與能源工程學系：工程數學#146107

1. Find the particular solution of the following initial value problem. 

\( y'' + 2y' + 3y = 0 \) with y(0) = 1 and y ′ (0) = −5

Then Find lim \( \lim_{t\to\infty} y(t). \)

2. Find the complete solution of the 2^nd-order non-homogeneous ODE:

\( y'' - 2y' + y = x^2 - x + 3 \)

3. Find the inverse Laplace transform of the following function F(s): 

\( F(s) = \frac{s^2}{(s+1)(s^2+4)} \)

4. If \( A] = \begin{bmatrix} 3 & 0 & 0 \\ -3 & 4 & 9 \\ 0 & 0 & 3 \end{bmatrix} \), determine the eigenvalues of [A], find an eigenbasis 0 0 3 of [A], and conduct diagonalization of [A].