104年 - 104 國立嘉義大學_碩士班招生考試試題_電機工程學系：工程數學#145950

1. Determine the values of λ for which the following system of equations has nontrivial solutions. (10%)

2. The three vectors (3, 2, 0), (1, 5, −1), and (5, −1, 2) form a basis for R³. Use these vectors in the Gram-Schmidt process to construct an orthonormal basis for R³. (15%)

3. Consider the bases B = {(1, 0), (0, 1)} and B' = {(1, 2), (-1,-1)} of R². If u is a vector such that , find u_B.(10%)

(a) Find the Laplace transform of \( f(t) = t^2 u(t - 3) \). (5%)

(b) Find the Laplace transform of \( f(t) = t(\cos t + \sin t) \) . (5%)

(c) Find the inverse Laplace transform of \( F(s) = \frac{1}{(s - 2)(s - 3)(s - 6)} \). (5%)

6. Solve \( (\sin(xy) + xy\cos(xy) + ye^x) dx + (x^2\cos(xy) + e^x) dy = 0 \) . (10%)

7. Solve the differential equation \( y'' = 3 - (y')^2 \) (10%)

8. Solve for currents in the circuit of Fig. 1, assuming that the currents and charges are initially zero and that E(t) = 2u(t − 4) − u(t − 5), where u(t) is unit step function. (15%)