101年 - 101 國立嘉義大學_碩士班(甲組)招生考試試題_土木與水資源工程學系：工程力學#146240

1. Solve the ODE by using variation of parameters method, and \( y=e^x \) is one of the homogenous solutions. (20%) \[ 2xy'' + (1 - 4x)y' + (2x - 1)y = e^x \]

2. \( \sigma_{ij} = \begin{bmatrix} \sigma_{xx} & \tau_{xy} & \tau_{xz} \\ \tau_{yx} & \sigma_{yy} & \tau_{yz} \\ \tau_{zx} & \tau_{zy} & \sigma_{zz} \end{bmatrix} = \begin{bmatrix} 3 & 0 & -2 \\ 0 & 2 & 0 \\ -2 & 0 & 0 \end{bmatrix} \) is a stress tensor, find the principal stresses and the unit vectors of principal directions by using eigenvalues and eigenvectors. (20%)

(a) Calculate by direct integration of around the circular closed coutour of radius 1 centered at the origin.(10%)

(b) Calculate the same circulation as in (a) by use of Stokes’s theorem. (10%)

4. The function is define as Figure 1. Find its Laplace transform. (20%)

(a) \( y + e^{x} + xy' = 0 \) .. (10%)

(b)\[ x^2 y'' - 3xy' + 4y = 0 \] . (10%)