課程名稱︰工程數學-微分方程
課程性質︰系訂必修
課程教師︰林清富 張帆人 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）︰2009.01.14
考試時限（分鐘）：150分鐘
是否需發放獎勵金：是
試題 :
1.(5%)Find a power serier solution of the differential equation y"-4xy'-4y=e^x
about x=0.
2.(5%)Is x=o an ordinary or a singular point of the differential equation
xy"+(sin x)y=0?
3.(5%)If the Laplace transform of f(t) is F(S), and k>0, then find the Laplace
transform of e^at*f(t-k)U(t-k).
4.(10%)Use the Laplace transform of the initial-value problem ty"+y'+ty=0,
y(0)=1,y'(0)=0 to show that L{J0(t)}=1/√(s^2+1).(Hint:F(s)=L{F(t)},
f(0)=lim s→∞ sF(s) )
5.(5%)Use the Laplace transform to solve the following BVP y"+2y'+y=0,y'(0)=2,
y(1)=2.
-1
6.(5%)Find the following inverse Laplace transform L {8k^3s/(s^2+k^2)^3}.
┌ 0 1 0 ┐
7.(20%)Let A=│ 0 0 1 │,
└ 0 0 0 ┘
┌ 1 ┐
(a)Solve dX(t)/dt=AX(t) with initial condition X(0)=│ 1 │.
└ 1 ┘
(b)Find Φ(t), such that the solution of (a) is X(t)=Φ(t)X(0) where X(0) is
the initial condition at t=0.
(c)Find Φ(t), such that the solution of (a) is X(t)=Φ(t)X(2) where X(2) is
the initial condition at t=2.
┌ 0 ┐
(d)Solve dX(t)/dt=AX(t)+f(t) with initial condition X(0)=│ 0 │ and
└ 0 ┘
┌ 0 ┐
f(t)=│ 0 │.
└ sin(t) ┘
8.(15%)
(a)Show that the product of two odd functions is even.
T
(b)Show that ∫ f(t)=0 if f(t) is an odd function.
-T
(c)Let f(t)=e^-t for 0≦t≦1. Please sketch the plots of the associated half
range expansions of sine series, cosine and Fourier series for -2≦t≦2;
respectively.
9.(15%)
(a)(8)The wave equation a^2Exx=Ett, 0<x<L, t>0 is subject to the initial and
boundary conditions:
E(0,t)=E(L,t)=0, t>0
E(x,0)=f(x), Et(t=0)=0, 0<x<L
Please show that the solution of the wave equation can be written as
E(x,t)=0.5[f(x+at)+f(x-at)]
[Hint:use the identity 2sinαcosβ=sin(α+β)+sin(α-β)]
(b)(7)If the wave equation is define for the infinite region -∞<x<∞ and t>0
with the initial and boundary conditions removed, then please give a
procedure that will lead to the general solution of the wave equation with
the form
E(x)=F(x+at)+G(x-at)
where F and G are two arbitrary twice-differentiable function.
10.(15%)Use the result that Fourier transform of e^(-x^2/4p^2) is
2√πpe^(-p^2α^2) to solve the following equations
(a)(7)Ut=DUxx, -∞<x<∞, t>0
U(x,0)=δ(x), -∞<x<∞.
(b)(8)Ut=-U/τ-vUx+DUxx, -∞<x<∞ t>0
U(x,0)=δ(x), -∞<x<∞.
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