109 年 國立臺灣大學工程科學及海洋工程學系碩士班丙組《工程數學（L）》

1. First, verify that the given function is a solution of the given differential equation, for any constants A, B.
   Then, solve for A, B so that y satisfies the given initial or boundary conditions.
   $y'' + 4y = 8x^2$; $y(x) = 2x^2 - 1 + A\sin 2x + B\cos 2x$;
   $y(0) = 1$, $y'(0) = 0$

2. From the residue theorem of the complex integral calculus, that
   $\int_0^\infty \frac{x^{\alpha-1}}{1+x} dx = \frac{\pi}{\sin(\alpha \pi)}$
   for $0 < \alpha < 1$. Show that $\Gamma(\alpha)\Gamma(1-\alpha) = \frac{\pi}{\sin(\alpha\pi)}$ for $0 < \alpha < 1$.

HINT: $B(p,q) = \int_0^\infty \frac{t^{p-1}}{(1+t)^{p+q}} dt = \frac{\Gamma(p)\Gamma(q)}{\Gamma(p+q)}$ for $p>0, q>0$.

3. Decompose the given matrix as the sum of two matrices, one symmetric and one skew symmetric.

4. Let $\sigma$ be the mass density of a (negligibly thick) distribution of mass over a surface S. That is, $\sigma$ is the mass per unit area at each point on S; it may vary over S. Then the x, y, z coordinates of the center of gravity are defined as
   $x_c = \frac{1}{M} \iint_S x \sigma \, dA, \quad y_c = \frac{1}{M} \iint_S y \sigma \, dA, \quad z_c = \frac{1}{M} \iint_S z \sigma \, dA$
   where $M = \iint_S \sigma \, dA$ is the total mass.

Evaluate $M$ and $x_c$ in S is the spherical surface $x^2 + y^2 + z^2 = 9$; $\sigma = 4+x$.