推 Forsomewell:連這都有人要賺批幣.....我想聽強者解惑第8題啦...Q_Q 01/20 09:24
課程名稱︰工程數學-微分方程
課程性質︰系訂必修
課程教師︰黃天偉 丁建均 林清富 管傑雄
開課學院：電資學院
開課系所︰電機系
考試日期（年月日）︰2008.1.16
考試時限（分鐘）：10:20~12:30
是否需發放獎勵金：是，謝謝
（如未明確表示，則不予發放）
試題 :
1. (x^2)(y")+xy'+(x^2-1/4)y=0
(a) (10%) Find power series solution about x=0, then expressed in terms of
sin(x) and cos(x).
(b) (5%) Find the general solution in terms of spherical Bessel functions
on the interval (0,∞).
2. Suppose y(x) is a piecewise continuous function of x and of exponential
order. It follows the differential equation xy"+y'-4y=0.
(a) (10%) Please find the Laplace transform Y(s) of y(x).
(b) (8%) Expand Y(s) with power series of 1/s and then the y(x) expanded
with power series of x.
(c) (7%) How many independent solutions can you get for the differential
equation with the Laplace transform? Please give the reason.
3. Find the general solutions of
(a) (8%) ┌ dx/dt = -x-2y+1
│
└ dy/dt = 3x+4y
(b) (7%) ┌ 6x = 4(dx/dt)-(dy/dt)
│
└ 6y = 2(dx/dt)+(dy/dt)
4. Suppose that x(t)=1 and y(t)=e^t and the interval is tε[0,1].
(a) (5%) Find the norms of x(t) and y(t) if the weight function is w(t)=t.
(b) (5%) Find c such that x(t) and y(t)-cx(t) are orthogonal with respect to
the weight function w(t)=t on an onterval tε[0,1].
5. (7%) Use the method of separation of variables to solve x(a2u/axay)+u=0.
(PS. "a2u/axay" 表示u對x作偏微分後，再對y作偏微分)
6. (10%) Solve the ditterential equation (a2u/ax2)+(a2u/ay2)=0, 0<x<a, 0<y<b,
subject to the boundary conditions:
u(0,y)=F(y), u(a,y)=G(y) 0<y<b
u(x,0)=f(x), u(x,b)=g(x) 0<x<a
7. (10%) Solve that a solution of the BVP.
(a2u/ax2)+(a2u/ay2)=0, -∞< x <∞, 0<y<1
au/ay|(t=0)=0, u(x,1)=f(x), -∞< x <∞
is
∞ ∞ cosh(αy)cosα(t-x)
u(x,y)=(1/π)∫ ∫ f(t)--------------------dtdα
0 -∞ coshα
8.(8%) Use Fourier series to solve the differential equation x"+10x=f(t),
subject to the initial conditions x(0)=0, x'(0)=0, where
┌ 5 0<t<π
f(t)= │ ; f(t)=f(t+2π).
└-5 π<t<2π
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