台師大數學系
微分方程第一次期中考
(2015.10.21)
考試範圍︰
13; 21; 22; 24; 25; 26; 28
Total score (105 points)
︰
1
(20 points) (1) Determine the order of the given differential equations;
(2) state whether the equation is linear or nonlinear, and (3) state that the
equation is ordinary or partial differential equation.
(a)
u
t
+
uu
x
= 6 +
u
xxx
,
(b)
dy
dt
+
ty
3
= 0
,
(c)
u
xxxx
+
u
xyx
+
u
yyy
= 0
,
(d)
d
2
y
dt
2
+
sin
(
t
+
y
) =
cos
t
2
(15 points) Solve the initial value problem (IVP)
dy
dx
=
4
x
+ 2
2(
y

1)
, y
(0) =
3
2
,
and determine the interval in which the solution exists. If its initial value is
replaced by
1
2
, solve the IVP and its interval.

2.0

1.5

1.0

0.5
0.0
0.5
1.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Page 1
Total pages: 2
台師大數學系
微分方程第一次期中考
(2015.10.21)
3
(10 points) Consider the initial value problem
dy
dx

y
=
x
+
e
x
, y
(0) =
y
0
.
Find the value of
y
0
that separates solutions that grow positively as
x
→ ∞
from those that grow negatively. How does the solution that corresponds to
this critical value of
y
0
behave as
x
→ ∞
4
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