edo1

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  AP  = 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