Transcript ENGG2013 Lecture 23 - Chinese University of Hong Kong

ENGG2013 Unit 23
First-order
Differential Equations
Apr, 2011.
Yesterday
• Independent variable, dependent variable and
parameters
• Initial conditions
• General solution
• Direction field
• Autonomous differential equations.
– Phase line
• Equilibrium
– Stable and unstable
The initial value problem
• Given a differential equation,
A function in x’, x and t
and some initial condition x(0) = x0,
find a function x(t) satisfying the differential
equation and the initial condition.
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Discharging a capacitor through a
resistor
+
V0
i(t)
V’(t) = –V/(RC)
V(0) = V0.
Autonomous
first-order DE
• Initial voltage across the
capacitor is V0.
• The switch is closed at t = 0.
• Voltage drop is proportional to
electric charge in capacitor.
V(t) = Q(t) / C
V’(t) = i(t) / C
• Voltage drop at resistor is
directly proportional to
current.
V(t) R = i(t)
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Population model for bacteria
Autonomous
first-order DE
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http://en.wikipedia.org/wiki/Bacteria
• Bacteria reproduce by binary fission.
• The rate of change of the population P(t) is
proportional to the size of population:
dP/dt = k P
where k is a positive proportionality constant.
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Radioactive decay
Autonomous
first-order DE
Proportionality
constant
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http://en.wikipedia.org/wiki/Radioactive_decay
• Radioactive decay is the process
by which an atomic nucleus of an
unstable atom loses energy by emitting ionizing
particles.
• The number of decay events is proportional to
the number of atoms present.
• Let N(t) be the number of radioactive atom
at time t.
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Phase line for x’ = kx
• k>0
Unstable equilibrium
Phase line
x
0
x
• k<0
Phase line
Stable equilibrium
x
0
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x
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The parameter a in x’ = ax
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• Three types of behaviour
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a>0
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– a > 0, exponential growth
– a = 0, constant solution
– a < 0, exponential decrease
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10
5
0
0
1
2
3
4
5
6
7
8
9
5
5
4.5
4.5
a<0
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3.5
4
3.5
3
3
x
2.5
2
2
1.5
1.5
1
1
0.5
0
a=0
2.5
0.5
0
1
2
3
4
5
6
7
8
9
0
0
1
2
3
4
5
6
7
8
9
t
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General Solution to x’=kx
• Each function of the form f(t) = C ekt is a
solution to x’ = k x.
• Easy to verify
f’(t) = (C ekt)’ = C (ekt)’ = C (k ekt) = k f(t)
Deriving the general solution by
power series
• Suppose we do not know that exponential function is a
solution to x’=kx. We can derive it using power series
method.
• Suppose that the solution is a power series in the form
c0+c1t+c2t2+…, where c0, c1, c2 ,… are constants to be
determine.
• Assume that we can differentiate term-by-term
(c0+c1t+c2t2+c3t3+…)’ = c1 + 2c2t + 3c3t2 + …
• By comparing like term,
c1 + 2c2t + 3c3t2 + … = k(c0 + c1t + c2t2 + …)
–
–
–
–
c1 = k c0
c2 = k c1/2 = k2 c0/2
c3 = kc2/3 = k3 c0/3!
in general, cn = kn c0/n!
General solution to x’=kx
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Typical application
Iodine-131
• The half life of Iodine-131 is about 8 days.
• Suppose that is one Becquerel (Bq) of Iodine131 initially. Find the number of days until the
radioactivity level drop to 0.01 Bq.
• Let x(t) be the radioactivity level on day t.
• Initial condition x(0)=1.
Unknown parameter
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Solution
• General solution x(t) = C e– t.
– Need to determine unknown constants C and .
•
•
•
•
x(0) = 1  C=1.
x(8) = 0.5  0.5 = e–8  =0.0866.
Therefore, x(t) = e– 0.0866t.
Solve 0.01 = x(t) = e– 0.0866t.
 t  53 days.
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CLASSIFICATION OF FIRST-ORDER
DIFFERENTIAL EQUATIONS
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Nomenclatures
• “First-order”: only the first derivative is involved.
• “Autonomous”: the independent variable does
not appear in the DE
• “Linear”:
– “Homogeneous”
– “Non-homogeneous”
c(t) not identically zero
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Examples
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First-order
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Falling body with linear air friction
kv
Positive direction
• The air resistance is in
the direction opposite to
that of the motion. The
retarding force is directly
proportional to v,
where v stands for the
speed, and  is a
constant between 1 and
2.
– Slow speed: =1.
– High speed: =2.
Suppose speed is slow
and =1.
m
mg
g –10 m/s2
k>0
Linear non-homogeneous
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RL in series
Current
• Physical laws
– Voltage drop across resistor = VR(t) = R I(t)
– Voltage drop across inductor = VL(t) = L I’(t)
L
From Kirchhoff voltage law
VR(t) + VL(t) = 0.5 sin(wt)
0.5 sin (wt)
R
Linear non-homogeneous
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RC in series
• Physical laws
– Voltage drop across resistor = VR(t) = R I(t)
– Voltage drop across inductor = C VC(t) = Q(t)
Charge
Vc
C
From Kirchoff voltage law
VC(t) + VR(t) = sin(wt)
sin(wt)
R
Linear non-homogeneous
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