課程名稱︰偏微分方程式一
課程性質︰數學系選修
課程教師︰夏俊雄
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2016/12/22
考試時限（分鐘）：50
試題 :
1. (40 points) Solve the following equation:
\[\begin{cases}
x_1 u_{x_2} - x_2 u_{x_1} = u \text{ in }\Omega,\\
u = g \text{ on }\Gamma,
\end{cases}\]
where Ω is the quadrant $\{x_1 > 0, x_2 > 0\}$ and $\Gamma = \{x_1 > 0,
x_2 = 0\}$.
2. (40 points) Solve the following equation:
\[\begin{cases}
u_{x_1} + u_{x_2} = u^2 \text{ in } \Omega,\\
u = g \text{ on } \Gamma,
\end{cases}\]
where Ω is the half space $\{x_2 > 0\}$ and $\Gamma = \{x_2 = 0\}$.
3. (40 points) Solve the following equation:
\[\begin{cases}
uu_{x_1} + u_{x_2} = 1,\\
u(x_1, x_1) = \frac12 x_1.
\end{cases}\]