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Differential Equations

Their Origins

Dillon & Fadyn

Spring 2000

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In The Beginning

Newton invented differential equations to

describe physical laws.

Many of the general laws of nature are

best expressed as differential equations.

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Examples

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Newtons Second Law ofMotion

The force acting on a body is equal to

F =dt

d)(mv = m

dt

dv

the rate of change of

the momentum of the body.

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Population Growth

The Exponential Model

The rate

dt

d

at which a population grows

P

is directly proportional to

k=

the size of the population itself.

P

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Notice

In the exponential model,

is constant.

P

dtdP/ k=

the relativegrowth rategrowth rate

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Relative Growth Rate

Absolute Growth RateP

dtdP/

dtdP

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Population Growth

The Verhulst Model

The relative growth rate of a population

PdtdP/

but isa function

)(f=

is nota constant,

P

of the size of the population.

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In the Verhulst Model

PdtdP/ )(Pf=

dt

dPP)(Pf=

)(PfThe function

can assume various forms

leading to different models.

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The Logistic Model

PbaPf =)(

As the population gets larger,

the relative growth rate

A Version of The Verhulst Model

PdtdP/ a= P b

decreases.

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Logistic Model

a

a/b

PdtdP/ Pba =

P

PdtdP/

Relative growth rate as a function

of the population.

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The Predator-Prey Model

dx dt x a by

dy dt y c dx

/ ( )

/ ( )

=

= +

Population Growth

A nonlinear System of D.E.s

Youre all experts on this now, yes?

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Compare Relative GrowthRates

LogisticLogisticP

dtdP/ a= Pb

ExponentialExponentialP

dtdP/k=

dx dtx

dy dt

y

a by

c dx

/

/

=

= +

Predator-Prey

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More Examples

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LRC Circuits

E(t)

R for resistance

C for capacitance

Electromotive force (battery)

Lfor inductance

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Kirchoffs Law

In words:

The sum of the voltagedrops across the passive elements

in the circuit equals the applied voltage.

Passive elements:

Applied voltage:

inductor, resistor, capacitor

what the battery supplies

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Voltage Drops?

Across the inductor dtdIL /

Across the resistor IR

Across the capacitor QC )/1(

HereI is current, Q is the chargeon the capacitor.

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Special Notes

The current is the same at all points inthe simple circuit we have here.

The capacitor is the only element with acharge associated to it.

The current is the first derivative of the

charge on the capacitor.

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The Model

A differential equation that describesthe relationships in the circuit

QRILtECdt

dI 1)( ++=

Independent variable t

Dependent variable I , Q

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Theres A Problem

Two dependent variables are o.k. for apartial d.e. or for a system.

This model should only have onedependent variable.

Use dtdQI /= to fix the problem.

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Substituting

QRILtECdt

dI 1)( ++=

QRLtEC1)( ++=

dt

dQI =

dt

dQ

2

2

dt

Qd

dtdI

=

2

2

dt

Qd

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The Model for an LRC Circuit

QRLtE C1)( ++= dtdQ

2

2

dt

Qd

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Spring-Mass Systems

Imagine a mass m suspended froma spring with a fixed support.

Suppose the whole system is in a dampingmedium, like air, or water, or jello.

Suppose further that there is a drivingforce,f(t), making the mass oscillate.

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The Model

kxcmtf dtdxdtdx ++= 22

)( x is the displacement of the mass,

measured from the resting position c is a constant depending on the

damping medium

kis the so-called spring constant (fromHookes Law)

tis the independent,x the dependent

variable

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Compare

QRLtE C1

)(++=

dt

dQ2

2

dt

Qd

kxcmtfdtdx

dt

dx++= 2

2

)(

LRC Circuit Model

Spring-Mass System Model

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One Model

Two entirely different

applications

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Final Remarks

We still havent solved a differentialequation, but now we know what they

might be good for.