[試題] 105上 陳俊全 常微分方程導論 期中考
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）︰105.11.11
考試時限（分鐘）：2 hours
試題 :
Part I.
1. (40 pts) Solve the following equations.
e^(x-y)
(1) dy = --------- dx.
y + 3
√y √x
(2) (----- + x)dx + (----- - y)dy = 0.
√x √y
x^3 + x(y+1)
(3) y' = --------------, y(1) = 1.
x^2 + 1
x^2 + y^2
(4) y' = -----------, y(2) = 1.
xy
y^2 + 1
(5) y' = --------------.
y^3 + y + xy
2. (20 pts) Solve the following equations.
(1) y'' + 6y' + 9y = 0, y(1) = 1, y'(1) = -1.
(2) y'' + 2y' + 3y = sin t.
Part II. Choose 2 from the following 3 problems
3. (20 pts) Let T(t) satisfy Stefan's law of cooling
dT 4 4
---- = a - T , T(0) = 1,
dt
where a is a constant greater than 1.
(1) Show that T(t) is increasing for t ＞ 0 and lim T(t) = a.
t→∞
dy 3
(2) Let y(t) satisfy ---- = 4a (a-y), y(0) = 1. Show that y(t) ＞ T(t) for
dt
t ＞ 0.
4. (20 pts) Let
t
ψ(t) = 0, ψ (t) = ∫ sψ(s)ds + 3, n = 0, 1, 2, 3,...
0 n+1 0 n
(a) Show that 0 ≦ ψ(t) ≦ 6 for 0 ≦ t ≦ 1.
n
(b) Show that lim ψ(t) exists for 0 ≦ t ≦ 1.
n→∞ n
(c) Find lim ψ(t) for 0 ≦ t ≦ 1.
n→∞ n
5. (20 pts) Show that if
∂N ∂M
----- - -----
∂x ∂y
--------------- = H(xy)
xM - yN
for some function H, then the equation M(x, y)dx + N(x, y) dy = 0 has an
integrating factor of the form μ(xy). Give the general formula for μ(xy).
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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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