M351 Study Guide 1 (S. Zhang)
2
1. (2.1:22)
Find the critical points and phase portrait.
Classfy each critical point Sketch all typical solution curves
on the graph of direction eld.
y(x)
1
y = y2 y3.
ans:
K
0
1
y = y 2 y 3 = y 2 (1 y )
By y = 0, we have
This is RREF. x1 , x2 , x4 are free.
M351 H6 (S. Zhang) 8.3-4
1. (7.6:a1)
Let x1 = c1 , x2 = c2 and x4 = c3 .
Show S is not a subspace
S = cfw_f (x) = a + bx + cx2 P2 | f (2) = f (1)2
x1
c1
x2 c2
x = x3 = 1 + c3
x4 c3
x5
0
1
0
0
0
0
1
0 0
Solve it as a rst order linear equation:
M351 H2 (S. Zhang) 2.7,2.8,3.1
0. 2.7: 13-14, 21-25, 31, 33
A + 0.02A = 4
1. (2.7:13)
A thermometer is removed from a room where
the temprature is 70 degrees. And is taken outside where
the air termperature is 10 d
M351 Study Guide 1 (S. Zhang)
9. (2.8:2)
The number N (t) of people who are exposed to a
particular advertisement is govened by the logistic equation.
Initially N (0) = 500, and it is observed that N (1) = 1000.
Solve for N (t) if it is predicted the limi
M351 H1 (S. Zhang) 2.1 2.2 2.3
2
0. 2.1: 1-2, 19-22, 29
1. (2.1:22)
Find the critical points and phase portrait.
Classfy each critical point Sketch all typical solution curves
on the graph of direction eld.
y(x)
1
y = y2 y3.
K
ans:
0
1
1
2
3
4
5
x
y = y 2
M351 H4 (S. Zhang) 3.5, 3.8
satises eqautions:
0. 3.5: 1-2, 11-12, 23, 25-26
u1 y1 + u2 y2 = 0
1. (3.5:12)
u1 y1 + u2 y2 = f
Find the general solution
The solutions are
y 2y + y = et /(1 + t2 )
0
f
u1 =
ans:
For yc = yH , r2 2r + 1 = 0,
0
tan x
=
(r 1)2 =
M351 Examples for 7.6, 8.2 (S. Zhang)
1. (Quiz5)
(3) (The system is in RREF. We view the system as
x2
(1) Show S is not a subspace
=0
x5 = 1.
x1
S = cfw_x2 | x1 + 2x2 = 3
x3
When looking from left, we see x2 and x5 . These two variables are not free. The
c1
M351 H5 (S. Zhang) 7.6,8.1,8.2
2c3
c1
0. 7.2: 44-46, 51-52 7.3: 23-24, 41-42 7.6: 11-20, 23-28
=0
+ 5c3
=0
+ c2
Let c3 = 1,
1. (7.6:12)
Determine the following is a subspaces of
C (, ):
c1 = 2
c2 = 3
c3 = 1
S = cfw_f | f (0) = 1
ans: No, not a subspac