課程名稱︰常微分方程導論 課程性質︰數學系必修 課程教師︰陳俊全教授 開課學院：理學院 開課系所︰數學系 考試日期（年月日）︰106.1.13 考試時限（分鐘）：2 hours 試題 : Choose 4 from the following 6 problems. 1. Solve the equation y''' - y'' - y' + y = cos t. ╭ a (t) a (t)╮ 2. Let A = │ 11 12 │. Consider the equation x'(t) = Ax(t). ╰ a (t) a (t)╯ 21 22 (a) What is the Wronskian W = W[y(t), z(t)] for the two column functions y(t), 2 z(t) in R ? (b) Show that if y(t), z(t) are solutions of x'(t) = Ax(t), then W' = (a (t) + a (t))W 11 22 and W either is identically zero or else never vanished in an interval. (c) Show that if W is nonzero at a point, then y(t), z(t) are linearly independent. 3. 2 (a) Let x(t) satisfies the equation x'' + 4x(x - 1) = 0. 1 2 2 2 Show that E(t) = ---x'(t) + (x(t) - 1) is a constant function of t. 2 (b) Show that if (x(t), y(t)) satisfies the system ╭ x' = y │ 2 ╰ y' = -4x(x -1), then x(t) satisfies the equation in (a). Find all critical points of the system and show that (1,0) is a stable critical point. (Hint: find a Lyapunov function V(x, y), or use the almost linear system.) ╭ 2 0 0 ╮ ╭ 2 1 1 ╮ │ │ │ │ 4. Let A = │ 0 1 0 │ and │ 0 1 3 │. │ │ │ │ ╰ 0 0 1 ╯ ╰ 0 0 1 ╯ At Bt (a) Find e ; (b) Find e . 5. Solve the linear system ╭ 1 1 ╮ ╭ 0 ╮ ╭ 2 ╮ x'(t) = │ │x(t) + │ │, x(0) = │ │. ╰ -2 3 ╯ ╰ t ╯ ╰ 0 ╯ 6. Use the Laplace transform to solve the equation y'' + y = sin t, y(0) = 1 = y'(0). -- 正妹也不過就是一組物質波方程式的特解罷了 --

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