Unformatted text preview: t)] + Ce
)e t/10 .
+ 400⇡ ⇡
1 + 400⇡
1 + 400⇡ 2
Longterm behaviour: as t ! 1, the last term ! 0, thus
50
50
IC: p(0) = p0 ) p0 = 150 1+400 2 + 50 ) C = p0 150 + 1+400 2
C
p(t) ! 150 50
[cos(2⇡ t) + 20⇡ sin(2⇡ t)] .
50
1 + 400⇡ 2 ⇡ t) + 20⇡ sin(2⇡ t)]+(p 150+
Solution of IVP: p(t) = 150
[cos(2
)e t/10 .
0
2
1 + 400⇡ equilibrium solutions.
1 + 400⇡ 2
By inspection of the DE there are no
Longterm behaviour: as t ! 1, the last term ! 0, thus
50
p(t) ! 150
[cos(2⇡ t) + 20⇡ sin(2⇡ t)] .
1 + 400⇡ 2
By inspection of the DE there are no equilibrium solutions.
5. (a) Integration by parts: let u = cos(2⇡ t) and dv = et/10 :
Z et/10 cos(2⇡ t) dt = 10et/10 cos(2⇡ t) + 20⇡ Z et/10 sin(2⇡ t) dt = 10et/10 cos(2⇡ t) + 200⇡ et/10 sin(2⇡ t)
Integration by parts: let u = cos(2⇡ t) and dv = et/10 :
Z Z 400⇡ 2 Z et/10 cos(2⇡ t) dt t/10
t/
cos(2 20⇡ et/ e sin(2⇡ t) dt
Thus t/10t/10 cos(2⇡ t) dt = 10e10 102cos(2⇡ t) + ⇡ t) + 20⇡10t/10 sin(2⇡ t)].
e e cos(2⇡ t) dt = 1+400 [e
dp
Z
(b) (Suppose S = (10 p)(1 + cos(2⇡ t)) 5(1 + cos(2⇡an = (5 p)(1 +2cos(2⇡ tD p(0) = p0 .
= D that the demand 10
b)
t/ also ﬂuctuates on10 )) annual basis, i.e., ), =
t/ t
et/10 cos(2⇡ t) dt
= 10e
cos(2⇡ t) + 200⇡ e
sin(2⇡ t) 400⇡
dt
(10 be )(1 + cos(2⇡ t))or Repeat part (a) with this demand function.)
Can psolved as linear . separable DE. Linear approach is shown here.
10
Thus et/10 cos(2⇡ t) dt = 1+400 2 [et/10 cos(2⇡ t) + 20⇡ et/10 sin(2⇡ t)].
Solution:
3
dp
(b)
= D S = (10 p)(1 + cos(2⇡ t)) 5(1 + cos(2⇡ t)) = (5 p)(1 + cos(2⇡ t), p(0) = p0 .
dt
Can be solved as linear or separable DE. Linear approach is shown here. 3 AMATH 350 Page 6 Assignment #3  Fall 2013 dp
+ (1 + cos(2⇡ t))p = 5(1 + cos(2⇡ t)).
dt
1
Integrating Factor: e 1+cos(2 t) dt = et+ 2 sin(2 t) .
Standard form: 1 et+ 2 sin(2 t) dp dt 1 + (1 + cos(2⇡ t))et+ 2 sin(2 t) 1 p = 5(1 + cos(2⇡ t))et+ 2 1
d t+ 1 sin(2 t)
e2
p = 5(1 + cos(2⇡ t))et+ 2
dt
1
1
et+ 2 sin(2 t) p = 5et+ 2 sin(2...
View
Full Document
 Fall '11
 Noidea
 Cos, Boundary value problem, dt

Click to edit the document details