課程名稱︰常微分方程導論
課程性質︰必修
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）︰2012/09/18
考試時限（分鐘）：
試題 :
ODE QUIZ 3 9/18 2012
You have to turn in the first two problem sets in class.
1. Solve the following differential equations.
x''(t) - x'(t) - 2x(t) = 0, x'(0) = 1, x(0) = 0.
2x''(t) - x'(t) - 3x(t) = 2t, x'(0) = 2, x(0) = 1.
x''(t) - 4x'(t) + 4x(t) = t^2 + 2t +1, x'(0) = 0, x(0) = 1.
x''(t) + tx'(t) + 2x(t) = 0, x'(0) = 1, x(0) = 2.
2. Solve the following differential equation and find the maximal interval of
existence.
y'(t) = 2y(1 - y/10), y(0) = 12.
y'(t) = 5y(1 - y/12), y(0) = 5.
3. State and prove the contraction mapping theorem for contraction mappings on
R^n space.
4.State and prove the Ascoli-Arzela theorem.