101年 - 101 國立嘉義大學_碩士班(乙組)招生考試試題_資訊工程工程學系：工程數學#146219

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

2. Solve \( x^2 y'' - 4xy' + 6y = 7 \ln(x^2) \) .(12%)

(a) the Laplace transform of \( f(t) = e^{-5t}(t^4 + 2t^2 + t) \); (7%)

(b) the Laplace transform of \( \int_{0}^{t} e^{\tau} (t - \tau) d\tau \) .(7%)

4. Solve \( y'' + 9y = r(t), \), where \( r(t) = 8 \sin t [u(t) - u(t - \pi)] ; y(0) = 0, y'(0) = 4. \) (12%)

5. Solve Laplace’s equation \( \nabla^2 u = 0 \) for a rectangular plate subject to the givenboundary conditions \( \begin{cases} u(0, y) = 0 & u(2, y) = 100 \\ u(x, 0) = 0 & u(x, 1) = 200 \sin 4\pi x \end{cases} \) .(12%)

6. Evaluate \( \oint_{C} \frac{z^{2}+1}{(z-1)^{2}(z+2i)} dz \) , where the contour C is the circle\( |z+i|=7 \) .(10%)

(a) Find the inverse of [A] . (8%)

(b) Find all the eigenvalues and their corresponding eigenvectors of [B]. (10%)

8. Consider the data (2, 1), (3, 2), (4, 3), (5, 2) on the 2D plane. Find the least squaresline for the given data. (12%)