109年 - 109 國立嘉義大學_碩士班招生考試試題_機械與能源工程學系：工程數學#146123

1.(ODE)

Find the particular solution of the following initial value problem.
\( y'' + 1y' - 6y = 0 \) with \( y(0) = 5 \) and \( y'(0) = -5 \) (25%)

2. Find the complete solution of the 2^nd-order non-homogeneous ODE: \[ y'' - 2y' + y = x^2 - x + 3 \quad (25\%) \]

3. (Laplace Transform) Find the inverse Laplace transform of the following function \( F(s) \): (25%) \[ F(s) = \frac{s^2}{(s + 1)(s^2 + 4)} \]

4. (Linear Algebra) If \( [A] = \begin{bmatrix} 3 & 0 & 0 \\ -3 & 4 & 9 \\ 0 & 0 & 3 \end{bmatrix} \), determine the eigenvalues of \( [A] \), find an eigenbasis of \( [A] \), and conduct diagonalization of \( [A] \). (25%)