ENGG2013 Unit 23

First-orderDifferential 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,

and some initial condition x(0) = x0,

find a function x(t) satisfying the differential equation and the initial condition.

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A function in x’, x and t

Discharging a capacitor through a resistor

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• 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) / CV’(t) = i(t) / C

• Voltage drop at resistor is directly proportional to current.

V(t) R = i(t)

V0

+

i(t)

V’(t) = –V/(RC)V(0) = V0.

Autonomous first-order DE

Population model for bacteria

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• 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.

Autonomous first-order DE

http://en.wikipedia.org/w

iki/Bacteri

a

Radioactive decay• Radioactive decay is the process

by which an atomic nucleus of anunstable atom loses energy by emitting ionizing particles.

• The number of decay events is proportional tothe number of atoms present.

• Let N(t) be the number of radioactive atomat time t.

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http://en.wikipedia.org/w

iki/Radioactive_deca

y

Proportionalityconstant

Autonomous first-order DE

Phase line for x’ = kx

• k>0

• k<0

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x

Phase line

0 x

Unstable equilibrium

x

Phase line

0x

Stable equilibrium

The parameter a in x’ = ax• Three types of behaviour

– a > 0, exponential growth– a = 0, constant solution– a < 0, exponential decrease

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0 1 2 3 4 5 6 7 8 90

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

t

x a=0

0 1 2 3 4 5 6 7 8 90

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

a<0

0 1 2 3 4 5 6 7 8 90

5

10

15

20

25

30

35

a>0

General Solution to x’=kx

• Each function of the form f(t) = C ekt is a solution to x’ = k x.

• Easy to verifyf’(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!

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General solution to x’=kx

Iodine-131

• The half life of Iodine-131 is about 8 days.• Suppose that is one Becquerel (Bq) of Iodine-

131 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.

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Typical application

Unknown parameter

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

Autonomous Linear

Homogeneous

Non-homogeneous

Falling body with linear air friction• 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.

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m

Pos

itive

di

rect

ion

mg

kv

Suppose speed is slowand =1.

Linear non-homogeneous

g –10 m/s2

k>0

RL in series

• Physical laws– Voltage drop across resistor = VR(t) = R I(t)

– Voltage drop across inductor = VL(t) = L I’(t)

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R

0.5 sin (wt)

L

Linear non-homogeneous

Current

From Kirchhoff voltage lawVR(t) + VL(t) = 0.5 sin(wt)

RC in series

• Physical laws– Voltage drop across resistor = VR(t) = R I(t)

– Voltage drop across inductor = C VC(t) = Q(t)

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Charge

R

C

sin(wt)

Vc

From Kirchoff voltage lawVC(t) + VR(t) = sin(wt)

Linear non-homogeneous