89 下 工數(二) 第一次期中考 90/3/29
1. Find
+
+ −
− s
e s
s
L s 3
2 1
) 1 (
3 (12%)
Hint : L
[
δ(t−t0)]
=e−st0 ,[
( 0)]
1e st0 t st H
L − = −
2. For the following 1st order O.D.E
3 2 3
4 3 2 d
d
+ +
−
−
= +
y x
y x x
y , (2)
use the method specified below to solve the general solution.
(No credit for other methods.)
(a) Use a transformation, (x,y)→ (X,Y), so that equation (2) becomes a homo- geneous equation. Then solve this homogeneous equation with Y= vX (9%) (b) Solve equation (2) as an exact equation (if not exact, find the integrating factor).
If the solution passes (x, y) = (1, 1), write down the specific solution. (12%)
3. Find the general solutions for the following ODEs :
(a) 2 2
d d x
y + 9y = x cos x
(16%)
(b) x
y x
y x
y
d d d
d d
d
2 2 3
3 + + + y = sin 2x + cos 3x (17%)
4. For the boundary value problem (20%) x x
y 2 d
d
2
2 = , y(0) = 2, y(1) = 0,
(a) Formulate the Green’s function G(x, z) :
(a1) governing equation (a2) boundary condition (a3) jump condition (a4) continuity condition (b) Find G(x, z)
(c) Write the solution y(x) in terms of the Green’s function G(x, z).
5. Solve the initial value problem (14%) e t
t y t
y y t y t
y y = −
+
+
+
2 2
2 3
3
d d d
d d
3d d
d ,
y(0) = 1, (0) 0
d ) d 0 d ( d
2
2 =
= t
y t
y ,
for y(t).
