106年 - 106 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#146409

1. The following system of equations has unique solutions. Solve this system using the method of Gauss-Jordan elimination with matrices. (10%)

2. Use the matrix inverse method to solve the following system of equations. (10%) \[ \begin{aligned} x_1 + 3x_2 - 2x_3 &= 1 \\ 2x_1 + 5x_2 - 3x_3 &= 5 \\ -3x_1 + 2x_2 - 4x_3 &= 7 \end{aligned} \]

3. The following vectors, (1, 1, 1), (2, 0, 1), (2, 4, 5), form a basis for R3. Use these vectors in the Gram-Schmidt process to construct an orthonormal basis for R3. (10%)

(a) Find the transition matrix P from the basis B = {(1, 2), (3, 0)} of R² to the basis B' = {(2, 1), (3, 2)}. (5%)

(b) Using the result obtained in 4(a). If , find (5%)

5. Solve \( y'' - 2y' + 10y = 20x^2 + 2x - 8 \). (10%)

6. Solve the given differential equation \( (y^2 + xy^3)dx + (5y^2 - xy + y^3 \sin y)dy = 0 \) (10%)

7. Solve \( x \frac{dy}{dx} + y = x^2 y^2 \) (10%)

8. Solve \( x^{2}y'' - 5xy' + 8y = 2\ln x \). (10%)

(a) Solve the integral equation \( f(t) = \cos t + \int_{0}^{t} e^{-\tau} f(t - \tau) d\tau \) (5%)

(b) Find the inverse Laplace transform of \( F(s) = \frac{s² + 6s + 9}{(s - 1)(s + 2)(s + 4)} \). (5%)

10. Find the Fourier series of the periodic function f(t) given in Fig. 1. (10%)