[試題] 98上 微分方程 期中考
課程名稱︰微分方程
課程性質︰大二上必修
課程教師︰黃天偉
開課學院：電機資訊學院
開課系所︰電機工程學系
考試日期（年月日）︰2009/11/18
考試時限（分鐘）：10:20~12:30 (130分鐘)
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
1.Solve the following 1st order equations (show the explicit solutions).
dy y^2 - 1
(a) (9%) ── = ────
dx 2x
dy
(b) (9%) ── = x + y
dx
dy
(c) (8%) ── = y + 2y^3
dx
dy x + 2y
(d) (9%) ── = - ────
dx 2x + 3
2.(7%) Please solve the differential equation of RC network and express the
rise time, Tr, of the output voltage, Vo, in terms of the RC time constant,
τ= RC. (The rise time, Tr, is defined as the time differences between 10%
to 90% of final voltage level.)
↑ ↑ __
1╔══ ┌─────┐ ┌───────┐ │ ╱
─═╝── ＋┼/\/\/\/\/\┼─┼──────┐│＋ │/
Vi －│R (Driver)│ │C (Receiver)＝│ Vo ￣￣￣
↓└─────┘ └──────↓┘－
3.(3%) A cake's temperature is expressed in (1). Based on the value in Table
(I), how long will it for this cake to cool off to a room temperaturn of
T-room? (Within Thermometer's measurement accuracy 0.5° ,expres the cooling
time in terms of the time constant, τ.)
T(t) ＝ T-room ＋ 200e^(-t/τ) (1)
Table (I)
┌───┬────┬────┬────┬────┬────┬────┬────
│ │ e^(-1) │ e^(-2) │ e^(-3) │ e^(-4) │ e^(-5) │ e^(-6) │ e^(-7)
├───┼────┼────┼────┼────┼────┼────┼────
│Value │0.367879│0.135335│0.049787│0.018316│0.006738│0.002479│0.000912
└───┴────┴────┴────┴────┴────┴────┴────
4.(7%) Find solution of xy'' = y' + (y')^3 .
5.(8%) Please solve the system of differential equations.
dx
── = 2x + 3y + 1
dt
dy
── = -x - 2y + 4
dt
6.(7%) Solve the differential euation (x^2)y'' + 4xy' + 2y = x.
7.(8%) Find the complementary function yc and the particular solution yp of
the differential equation (x^2)y'' - 2xy' + 2y = x
8. x'' + (ω^2)x = cos(γt), x(0) = x'(0) = 0
(a)(12%)
Show that when the difference between ω and γ is small, an approxima-
te solution,‘beats phenomenon’, is
1 sin(γt)
x(t) ＝ ─- A(t)─────
2 γ
where A(t) is function of t, ω, and γ. Please also find A(t).
(b)(3%)
When the ω= 2π x 90.9MHz and γ= 2π x 90.924MHz, find the‘envelop’
frequency of the approximate solution in (a).
9.(10%) Please solve the differential equation
y'' - 2y' + y = (e^t)arctan(t) + e^t.
--
※ 發信站: 批踢踢實業坊(ptt.cc)
◆ From: 140.112.244.32
推
11/24 02:25, , 1F
11/24 02:25, 1F