[試題] 105上 陳俊全 常微分方程導論 期末考
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).
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正妹也不過就是一組物質波方程式的特解罷了
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