课程名称︰工程数学-微分方程
课程性质︰必修
课程教师︰管杰雄
开课学院：电资学院
开课系所︰电机系
考试日期（年月日）︰2015/1/14
考试时限（分钟）：170
试题 :
1. (20%) Use Laplace transform to solve the following differential equation:
ty'' - 2y' + ty = 0. Subject to y(0) = A, a constant. y(t) = ?
2. (8%) Evaluate L^(-1){(1/s^2)/[1-e^(-s)]}. Hint: expand 1/[1-e^(-s)] as the
series of e^(-ns) and plot the picture.
3. (12%) Expand f(x) = x, 0 < x < L,
(a) in a cosine series,
(b) in a sine series, and
(c) in a Fourier series.
/ 14 12 10 \
4.(10%) Solve X' = (1/4)| -3 2 -5 | X
\ 0 0 8 /
5.(10%) Solve the following system of differential equations:
/ (D+1)x + (D+1)y = e^(-t)
\ 2Dx + (2D+1)y = t
6.(20%) Solve the following differential equation
k d^2u/dx^2 = du/dt ,for 0 < x < L, t > 0
(都是偏微分)
which subjects to the given boundary conditions:
ux(0,t) = ux(L,t) = 0 and u(x,0) = f(x) for 0 < x < L.
7.(20%) Use Fourier Integral method to solve Laplace's equation
d^2u/dx^2 + d^2u/dy^2 = 0 , for 0 < x < L, y > 0
(都是偏微分)
which subjects to the given boundary conditions:
u(0,y) = u(L,y) = 0 for y > 0 and u(x,0) = V, a constant, for 0 < x < L,
with the following expansions:
(a) u(x,0) = V, for all x 属于 R
(b) u(x,0) = V, for 0 < x < L, and u(x,0) = -V, for -L < x < 0, and u(x,0)
is periodical with the period of 2L.
Please solve the equation respectively with the above expansions.
(c) 加分题，抄在黑板上，大意是比较(a) (b) 的解，
讨论有限范围与全域的解唯一性。