[試題] 98上 電機系微分方程 期末考
課程名稱︰工程數學-微分方程
課程性質︰大二上必修
課程教師︰黃天偉 林清富 丁建均
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）︰2010/01/13
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.(10%) Find the power series solution of the following wquation about x=0.
1 2 ,, ,
──(1-x )y - xy + 3y =0
2
2.(8%) Find the Laplace transform of e^(-t)cosh(3t) + δ(t-2)
1
3.(7%) Find the inverse Laplace transform of ───────
(s-1)(s+1)s
4.(10%) Solve x(t) by the Laplace transform
d^3 d^2 d
───x(t) + 2───x(t) + 3──x(t) + 2x(t) = 0
dt^3 dt^2 dt
5. (a)(3%) Please write down the 1-D wave equation.
(b)(7%) Please find the general solution of the 1-D wave equation.
6. (a)(3%) Please write down the 2-D Laplace's equation with the two independent
variables represented by x and y, respectively, and the dependent
variable represented by u(x,y).
(b)(10%)Please solve the above 2-D Laplace's equation with the boundary
conditions:
u(0,y)=0, u(a,y)=G(y) for 0<y<b
u(x,0)=0, u(x,b)=g(x) for 0<x<a
-(x^2/4p^2)
7.(7%) Use the result that the Fourier transform of e is
-(p^2)(α^2)
2(√π)pe to solve the following equation:
φu u (φ^2)u
── = - ── + D──── -∞ < x < ∞ t>0. (註：φ是partial偏微分)
φt τ φx^2
u(x,0) = δ(x) -∞ < x < ∞
8.(5%) If y1 and y2 are linearly independent solution of the associated
homogeneous DE for y''+ P(x)y'+ Q(x)y = f(x), show in the case of a
-1
non-homogeneous linear second-order DE that Xp = Φ(t)∫Φ (t)F(t)dt
reduced to the form of
-y2f(x) y1f(x)
yp = y1∫────dt + y2∫────dt
W W
9.(10%) Solve the given linear system
, ╭ 1 2 ╮ ╭ 0 ╮
(a) X = │ │X + │ │
╰ -0.5 1 ╯ ╰ (e^t)tan(t) ╯
╭ 0 2 1 ╮
, │ │
(b) X = │ 1 1 -2 │X
│ │
╰ 2 2 -1 ╯
10.(3%) Please find and prove the smallest interval near the origin for the set
of functions is said to be orthogonal on the smallest interval:
(a) {cos(nx)}, n=1,3,5...
(b) {1,cos(nx)}, n=1,2,3...
(c) {1,cos(nx),sin(mx)}, n=1,2,3... m=1,2,3...
11. (a)(4%) Find the Fourier series of f(x) on the given interval
f(x) = 0, -π＜ x ＜ 0
= x^2, 0 ≦ x ＜ π
(b)(3%) Use the result of (a) to show
π^2 1 1
── = 1 + ── + ── + ...
8 3^2 5^2
12. (a)(5%) Find the c_n of the Complex Fourier Series of function f(x) defined
on an interval (-p,p)
∞
f(x) = Σ c_n e^(inπx/p)
n=-∞
(b)(5%) Use (a) to expand the following f(x) and prove c_n has the form of
sin(x)
Sinc function, ───
x
0, -(1/2) ＜ x ＜ -(1/4)
f(x)= 1, -(1/4) ＜ x ＜ 1/4
0, 1/4 ＜ x ＜ 1/2
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