作者dn890221 (车)
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标题[试题] 97上 杨馥菱 工程数学上 第2/2次期中考
时间Wed Dec 10 11:30:40 2008
课程名称︰工程数学上
课程性质︰系定必修
课程教师︰杨馥菱
开课学院:工学院
开课系所︰机械系
考试日期(年月日)︰971210
考试时限(分钟):65
是否需发放奖励金:是
试题 :
Reference: A two sided A4 size summary.
[ Series solution 40% ] 2
x
1. Find a series solution to y' + xy = e . If y(0) = -1, write out explicitly
the first 4 terms of your solution. (15%)
2
2. Solve the ODE: 3x y" + 4xy' - (3x+2)y = 0 around x = 0. Explain why you
choose Power series method or Frobenius method. Find
two linearly independent
series solutions for y(x) = C1y1(x) + C2y2(x). No need to determine C1, C2,
but solve for the indicial equation and the recursion formula. (25%)
[ Laplace Transform 35% ]
Use Laplace transform to solve the ODE. Provide all the details.
0 for 0≦t<4
3. y" + 4y = f(t) = { , where y(0) = 1 and y'(0) = 0. (20%)
3 for t≧4
4. y" + 4ty' - 4y = 0 subjected to y(0) = 0 and y'(0) = -7. (15%)
The initial value theorem lim y(t) = lim Y(s) may be of use.
t→0 s→∞
Please remember
L[tf(t)] = -F'(s)
[ Special function and Eigen function expansion 40% ]
5. Categorize the following Sturm-Liouville problem:
2
y" + λ y = 0 with y(0) - 2y'(0) = 0 and y'(1) = 0.
Determine the eigenvalue. How many eigenvalues are there?
d┌ dy┐ ┌ 4┐
6. Find the general solution to the ODE ─│x─│+│λx - ─│y = 0, where the
_ dx└ dx┘ └ x┘
transformation z = √λx will help. Now determine the solution to the
d┌ dy┐ ┌ 4┐ 1 1
following ODE: ─│x─│+│x - ─│y = 0, with y(0) = 0, y(-) = ─.
dx└ dx┘ └ x┘ 2 32
Please sketch your solution for x>0. The following information might be
useful when interpreting the boundary conditions:
At small x, 2
x x
J0(x) ~ 1, J1(x) ~ -, J2(x) ~ ─, ...
2 8
2 2 -1 4 -2
Y0(x) ~ ─ ln x, Y1(x) ~ -─ x , Y2(x) ~ -─x , ...
π π π
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