102年 - 102 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#146346

1. Find the general solution of the differential equation \( y''+5y'+6y=3e^{-2x}+e^{3x} \). (25%)

2. If \( L\{y\} = \frac{s+1}{s^2+s-6} \), what is y(t) ? (25%).

3. Calculate the work done by the force \( F = (2y - x^2 e^x)i + (\cos(4y^2) - x)j \) in moving a particle about the rectangular path \( C \) of the Figure below. (25%).

(a) Find the Fourier coefficient a₀ = ? (10%)

(b) Let \( f'(x) \) is continuous and \( f''(x) \) exists throughout the interval, find \( \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^2} \sin\left(\frac{n\pi}{2}\right) =? \) (15%)