L
dP =k a dt P = 100
⇒
P
− 100 = 0
dP =k a dt P =
P 3
⇒
P
P
−
3
√ ⇒
P
− −
P N
− 100)
P
−
P 3
− a√ P = 0
dP =k a dt a
P (P N
=0
dP =k a dt P = a P
−
P P N
−
P (P N
√ − a P ) P )
dM = P M + kM dt dM = kM dt M =
P =
P − M
k=
− √ √
dx = ax by x dt dy = cy x dt
x= y= x
t ax
y =0
a
−y√ x
√
dx = ax b(0) x dt dy = c(0) x dt
−
√
dy =0 dt dx = ax dt c0 y(t) y (t)
t 0
c0
≤ t ≤ 40
c0
limt→∞ 0,1 c0 y(t)
≤ ≤2 → 10c t→∞
c0 = 1
t t
→∞ (c) 0
(c) t 150 5 y 15 oc(t, 100/6, 60).
≤ ≤
t
− ≤ ≤
Rentrada = c0 (c0 q ) y y Rsalida = C 10 10 dy y = c0 dt 10
∗ ∗
−
dy y + dt 10 1 m+ =0 10 1 m= 10
−
t
yh = c1 e− 10
y p = A
y
0
c0 = 1,0
A = c0 10 A = 10c0
y = c1 e− t=0
t + 10c0 10
→ y = 10lb
0 + 10c0 10 10c0 = c1
10 = c1 e− 10
−
t + 10c0 10 t l´ım y = l´ım 10(1 c0 )e− + l´ım 10c0 t→+∞ t→+∞ 10 t→+∞ l´ım y = 0 + 10c0 y = 10(1
−
− c )e 0
−
t→+∞
l´ım y = 10c0
t→+∞
y = 10(1
−
− c )e 1
t
10
+ 10c1
y = 10
dy = dt
− 10y
dy y + =0 dt 10 1 m= 10
−
t
y = c1 e− 10 10
10 = c1 e− 10 1c1 = 10e 10
y = 10e e− 10
∗
t
y = 10e1− 10
y=
10
t 1− 10
10e
0 t
≤ t ≤ 10; ≥ 0.
t
y = c1 e− 10 t(10) y = 10 10
10 = c1 e1− 10 + 10c0 c1 = 10(1 y = 10(1
− c )e 0
−
− c )e 0
t
10
dy = dt
− 10y t
y = c1 e− 10
10 = c1 (1) t = 10
⇒ y = 10e
−
⇒c
= 10
⇒c
t
1
= 10e− 10 0
≤ t ≤ 10
1 = 3, 67 60
9, 956 = c1 e− 10 t = 10
1
⇒ y = 4016, 54e
−
70 10
⇒c
1
= 4016, 54
⇒ y = 4016, 54e
−
t
10
= 3, 66 120
9, 956 = c1 e− 10
⇒c
1
= 1, 62592x106
6
⇒ y = 1, 62592x10 e
−
t
10
= 3, 67
t
10e (3, 57 − 10c )e 4016, 5e y= 66 − 10c )e (3,1, 62592x10 e (0, 496 − 10c )e −
10
t 1− 10
0 t 1− 10
+ 10c0
t 7− 10
0 6
+ 10c0 10 + 10c 0 13− 10 + 10c0
−
0
t
t
0 10 60 70 120 130
≤ t ≤ 10; ≤ t ≤ 60 ≤ t ≤ 70 ≤ t ≤ 120 ≤ t ≤ 130 ≤ t ≤ 150;
.
t=2 t0 = 0
t = 25
t0 = 3 t
→ +∞
A(t) = cantidaddesolucin dA = (rapidezalaquelasustanciaentra ) dt dA dt
= R1
− (rapidezalaquelasustanciasale )
−R
2
(lb/min) R1 = QC
Q
C R1 = (3gal/min)(2lb/gal) R1 = 6lb/min
R2 = (3gal/min) R2 =
+
3 50
50
− 503 A
A=6 dA 3 + A =0 dt 50 dA 3 = A dt 50 50 A = 3dt 3 3t ln CA = 50
−
−
| |
CA = e
−3t 50
AH = K 1 e
−3t 50
AP = Z AP = 0 3Z =6 50 Z = 100
0+
3t
A(t) = K 1 e− 50 + 100 A(0) = 0 0 = K 1 e0 + 100 K 1 =
A(t) =
−98
−98e
−
3t 50
lb/gal
3 50
A
dA =6 dt
dA dt
A
+ 100
A(2) =
−
−98e
3(2) 50
+ 100
A = 12lb A(2) =
−
−98e
3(2) 50
+ 100
A = 12lb A(3) =
−
−98e
3(2) 50
+ 100
A = 85, 3lb e
m(t)
82◦ 90
94
◦
◦
180◦
t
65◦
→ +∞ dT = k(T T m ) dt dT = kdt (T T m ) ln(T T m ) = kt + c2
−
− − ln(T − T ) = kt + ln c T − T = c e m
m
2
kt
T = c2 e
k
C.It(0) = 82 F t(3) = 90◦ F t(6) = 94◦ F t(9) =? ◦
dT dt
= k(T
− T
m)
2
kt
+ T m
T
T (t) = c2 ekt + T m t(0)
⇒ T
m
= 82◦ F
T (t) = c2 ekt + 82 T (3) = c2 e3k + 82 T (6) = c2 e6k + 82 8 12 = 6k 3k e e
6k
⇒ 8e
3k
⇒8= c e ⇒ 12 = c e
= 12e3k
2
2
6k
⇒ k = 0, 13515
c2 = 5, 33 T = 5, 33e0,135155t + 82 dt = 9[s] T (9) = 100◦ F dT = k(T T m ) dt T (0) = 180◦ F
−
T (60) = 140◦ F T m = T habitacion = 65 ◦ F t =?
◦
→ T = 120, 90, 65 F
T = c2 ekt + T m = c2 ekt + 65 kt
T = c2 e
kt
+ T m = 115e
+ 65
⇐c
⇒k=
ln
2
= 115◦ F
15 23
60
T (t) = 115e−0,007124t + 65 T 65 = e−0,007124t 115
−
−65 ln T 115 0, 007124
t=
−
T = 120◦ F T = 90 ◦ F
⇒ t = 103, 54min
⇒ t = 214, 21min
=
−0, 007124
T = 65 ◦ F t = 1000
T H = T 0H
−c e
−
2
⇒ t = +∞
65◦ F
k t
t=0
→ T
H
0 = T 0H
= T 0H T A = T 0A
+c e
0A
2
(T 0 A
c2 =
k t
−
− T
e
−
−c e 2
k t
t=0
→ T
A
= T 0A
−e k t
− T H ) −e 0
k t
k t
k
e A −−T e H ) e − (T e A −−T e H ) e − (T 0
T H = T 0H
0
k − t
T A = T 0A
−
0
0
k − t
k t
k t
r 3
1g/cm t=0 mg = πr 2 hg πr 2 xg
k t
k t
h
ρ
≤ 0,5
x = x(t) t
xe = ρh g = 980cm/s2
p = 2π
ρh/g
ρ?0,5g/cm3 h = 200cm
p
peso
F = 0
− Empuje = 0
empuje = Peso πr 2 xe g = ρπr 2 hg xe = ρh
d2 x + ωx = 0 dt2 d2 x dt2
=
−ωx
x = A sin(ωt + φ) v = Aω cos(ωt + φ) a=
−Aω
2
sin(ωt + φ) =
2
−ω x
F = ma Peso
− Empuje = ma mg − πr gx = ma m(a − g) = −πr gx x a − g = −πr g m 2
2
2
m = ρπr 2 h a
a
−
a g 2 g = ddt2x
x g − g = −πr g rhoπr =− x h ρh 2
2
−
ω2 =
d2 x = dt2
− ρhg x
d2 x = dt2
−ρ x
g ρh
2
ω2 = ω=
g
T = T = p = 2π
ρh/g
2π
g ρh
g (ρh) ρh
2π ω = 2π
g
ρh
p = 2, 07min
M m m
F r =
r F r =
−GM m/r r
2
R = 3960
r M r
3
−GMmr/R
m t =0 r (t) =
2
−k r(t)
r t 2 k = GM/R3 = g/R
F =
−G mrm 1
2
2
F G F r =
3
−GMmr/R
r (t) = k 2 = g/R r (t) = 2 r (t) = g/R r2
−
−g/Rr(t)
2
−k r(t)
F = m dv dt
