101年 - 101 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#146248

1. Find the general solution of the given differential equation \[ y'' - 3y' + 2y = e^x . (25\%) \]

2. Calculate the work done by the force \( F = 5zi + xyj + x^2zk \) with the line integral \( \int_C F \cdot dr \) from point \( (0, 0, 0) \) to point \( (1, 1, 1) \) along the following two paths (a) straight line : \( x = y \), \( y = z \) and (b) parabolic curve : \( x = y \), \( z = y^2 \).(25%)

(a) Solve the equation given above from t = 0 to t. (15%)

(b) If the flow rate is \( \dot{Q}(t) = \frac{t}{R} \), solve the equation in part (a). (10%)

(a) Write down the equations of the problem for X and for T. (Solutions are not required) (10%)

(b) The solutions of the problem for \( X(x) \) and for \( T(t) \) are \[ X_n(x) = \sin\left(\frac{n\pi x}{L}\right) \]\[ T_n(t) = c_n e^{-n^2\pi^2 kt/L^2} \]