課程名稱︰微分方程
課程性質︰系定必修
課程教師︰張帆人 管傑雄 丁建均 黃天偉
開課學院：電資學院
開課系所︰電機系
考試日期（年月日）︰2011/1/12
考試時限（分鐘）：110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.(14 scores) Find the general solutions of
d
運算子(──)=D
dt
╭ D˙x(t)- D˙y(t)=0
│
(a) │
│
╰ D˙x(t)+ D˙y(t)=3[-x(t)+y(t)] (7 scores)
╭ D˙x1(t)- x1(t) = x2(t)
│ D˙x2(t) = x2(t)
(b) │ D˙x3(t) = x3(t)
│ D˙x4(t)- x4(t) = x5(t)
│ D˙x5(t) = x5(t) (7 scores)
╰
2.(16 scores)
(a)Find power series solutions about x=0.
"Also specify the region of convergence. "
(x^2 + 1)y"(x)-2y(x)=0
(8 scores)
(b)Find power series solutions about x=0.
(x^2)y"(x) + xy'(x) + (x^2-4/9)y(x)=0
(8 scores)
3.(10 scores) Legendre polynomials Pn(x) are orthogonal on the interval[-1,1].
The first 3 Legendre polynomials are
P0(x)=1
P1(x)=x
P2(x)=(3/2)x^2-(1/2)
(a) Verify {P0(x), P1(x), P2(x)} is an orthogonal set. (5 scores)
(b) Find the first three coefficients, a0, a1, a2 in the orthogonal series
expansion of f(x).
f(x)=a0P0(x)+ a1P1(x) +a2P2(x) +....
where f(x)=╭ -1, -1<x<0
│
╰ 1, 0<x<1 (5 scores)
4.(10 scores) Let Φn(x)≡e^(inπx/p), where i=√(-1), n are integers.
Define the set S as
S≡{...Φ-n(x), ...Φ-2(x),Φ-1(x),Φ0(x),Φ1(x),Φ2(x),...,Φn(x),...}
The inner product of Φm(x) and Φl(x) over [-p,p] is defined as
p
(Φm(x),Φl(x))=∫ Φm*(x)Φl(x)dx
-p
where Φm*(x) denotes the complex conjugate of Φm(x).
(a) Show that S is an orthogonal set. (5 scores)
(b) Assume f(x) is a piecewise continious real function over[-p,p].The
complex form Fourier series of f(x) is
∞
Σ Cn e^(inπx/p).
n=-∞
Show that 1 p
Cn=──∫ f(x)e^(inπx/p)dx.
2p -p (5 scores)
(φ^2)u(x,t) (φ^2)u(x,t)
5.(15 scores)Solve 16────── = ────── for π>x>0, t>0 with
φx^2 φt^2
(註：φ是partial偏微分)
╭ u(0,t)=0, u(π,t)=0
│ u(x,0)=sinx+2sin2x
│
│ φu(x,t)|
│ ────| =0
╰ φt |t=0
(a) Find u(x,t) by the method of separation of variables. (10 scores)
(b) Show that u(x,t) can be expressed as (1/2)[f(x+at) + f(x-at)]. (5 scores)
6.(5 scores)
(a) Expand f(x)=e^x, 0<x<1 in a cosine series. (3 scores)
(b) Find the value of f(999) in problem 6(a). (2 scores)
7.(10 scores) Using Fourier series to solve the following DE.
x" + 10x = f(t), x(0)=0, x'(0)=0
f(t)=╭5, 0<t<π f(t+2π)=f(t)
╰0, π<t<2π
8.(10 scores)
(a) Find the corresponding time-domain waveform f(t) of the S-domain function
e^s 1
──(───), and graph it.
s 1-e^-s (2 scores)
(b) FInd the corresponding time-domain waveform f(t) of the S-domain function
e^s 1
──(───), and graph it.
s 1+e^-s (2 scores)
(c) Use the f(t) of Problem 8(b) and Laplace Transform to solve the DE:
x"+ 2x'+ x = 5f(t) , x(0) = x'(0) = 0
(4 scores)
(d) Discuss the differences of responses from the Fourier Transform in
"Problem7" and the Laplace Transform in Problem 8(c). (2 scores)
9.(10 scores) Please use the following initial value problem
(t^2)y" + (t^2)y' +(t^2)y = 0, y(0) = 1, y'(0) = 0
to show the inverse Laplace Transform of 1/√(s^2+1) is Bessel Function:
-1 1
L {─────} =J0(t)
√(s^2+1)
--
※ 發信站: 批踢踢實業坊(ptt.cc)
◆ From: 140.112.4.185