Introduction to Differential Equations

● Introduction● What is a differential equation?● Malthusian growth and other examples

● Some applications● Radioactive decay

● Classification of differential equations● Summary

Introduction

● Differential equations frequently arise in modeling situations

● They describe population growth, chemical reactions, heat exchange, motion (all classical physics built on diff. eq's.), ... etc.

● Continuous analogue of discrete dynamical systems

What is a Differential Equation

● A differential equation is any equation of some unknown function that involves some derivative of the unknown function

● Classical example is Newton's law of motion● The mass of an object times its acceleration is

equal to the sum of the forces acting on it (“F=ma”)● Acceleration a is the first derivative of velocity

(a=d/dt v) and the second derivative of position (a=d2/dt2 x)

● F=m d2/dt2 x is an example of a diff. Eq's.

● More generally, diff. eq's. can be used to model

the temporal evolution of any system

Example: Malthusian Growth

● Discrete Malthusian growth, population Pn at

time n with growth rate r:

● Rearrange the above equation

● Note: change in population between the (n+1)st time and nth time is proportional to r times the size of the population at time n.

(E.g. a fraction r of individuals has an offspring)

Pn+1=Pn+r Pn

Pn+1−Pn=r Pn

... or Interest on Savings ...

● Could also understand

● Pn as the amount of savings we have at time n

● r as the interest rate

● The previous equation then just says that savings next year are equal to savings this year plus interest

Malthusian Growth (2)

● Let P(t) be the size of the population at time t● Assume that r is the rate of change per unit time per

individual in the population● Let ∆t be a small interval of time, then change in

population between t and t+∆t satisfies:

● The equation can be rearrange to give:

P(t+Δ t)−P(t )=Δ t rP (t)

P (t+Δ t)−P (t)Δ t

=rP(t )

Malthusian Growth (3)

● Discrete model was given by

● Left hand side should remind of the definition of a derivative. Take the limit of very small intervals of time ∆t -> 0:

● This is the continuous Malthusian growth model

in the form of a differential equation

P (t+Δ t)−P (t)Δ t

=rP(t )

limΔ t→ 0

P (t+Δ t)−P (t)Δ t

=dP (t)

dt=rP( t)

Solving Differential Equations

● Solving the discrete system

means finding a function Pn that satisfies the

above equation and meets some initial condition for P

0 and allows to answer the

question: “If we have population P0 at time zero

what is the size of the population Pn at time n?”

● This can be done easily ... seminar.

Pn+1=Pn+r Pn

Solving Differential Equations

● Solving

means finding a function x(t) that obeys the equation and meets some initial condition at time zero, e.g. x(0)=x

0.

● (So from some local rule how to progress in time from t to t+dt we find the global behaviour x(t) of the system)

dx (t)dt

=f ( x , t)

Slope fields ...

● dy(x)/dx is slope of y at position x. Hence f(y, x) defines a slope or direction field

● We look for a function

y(x) that is tangent to

the slope field at every

point

y

x

Slope field for dy/dx=x2-x-2 (from wikipedia)

Solving Malthusian Growth

● The differential equation for continuous Malthusian growth is

● Let c be an arbitrary constant and let's try a solution of the form

● Differentiating: i.e.

dP(t )dt

=rP (t) , P(0)=P0

P(t )=cer t

(*)

ddt

c er t=c r e

rt

ddt

P(t )=r P( t)

Solving Malthusian Growth (2)

● Hence solve the differential equation (*)● This is a family of solutions (parametrized by c), we

need to find the member of this family which obeys the initial condition P(0)=P

0.

● Hence the solution to the initial value problem is:

(e.g. exponential growth)

P(t )=cer t

P(t )=cer t→P(0)=cer 0=c=P0

→c=P0

P(t )=P0 er t

Example

● Consider the Malthusian growth model

with P(0)=100

● Find the solution and determine how long it takes for the population to double

dP(t )dt

=0.02 P(t )

Applications of Differential Equations

● Radioactive decay: Let R(t) be the amount of some radioactive substance at time t.● Radioactive substances transition into another state

at a rate k proportional to the amount of substance present

● The differential equation is:

● With initial condition R(0)=R0.

● In analogy to Malthusian growth:

dR( t)dt

=−kR (t )

R( t)=R0 e−k t

Harmonic Oscillator

● A Hooke's law spring exerts a force that is proportional to the displacement of the spring

● Newton's law of motion

F=ma=-cx● The simplest spring-mass

problem is

with

d2

dt 2 x+k2 x=0

k 2=c /m

A Swinging Pendulum

● A pendulum is a mass attached at one point that it swings freely under the influence of gravity ● Newton's law of motion

gives

● Where

d2

dt 2 Θ+ω2sin (Θ)=0

ω2=g/m

gravitationalconstant

mass of bob

Logistic Growth

● Most populations limited by food, space, or waste-build up and hence cannot grow according to Malthusian growth● Logistic growth model has a Malthusian growth

term and a term limiting growth due to crowding

● Where P is population size, r the (Malthusian) growth rate, and M the carrying capacity

● Will discuss solutions later

dPdt

=r P(1−P /M )

Lotka-Volterra Predator Prey Model

● Population of prey (x) and predator (y) interacting in an ecosystem

● System of differential equations for x and y

● No explicit solutions, but will discuss this system later

ddt

x=ax−bxy

ddt

y=−cy+dxy

Classification of Differential Equations

● Order of a differential equation● Is determined by the highest derivative in the

equation● E.g. Malthusian or logistic growth are 1st order,

Harmonic osci is 2nd order● Lotka-Volterra is a first order system of differential

equations

● Higher order equations can be transformed into systems of first order equations by introducing new variables

Transforming higher order ODEs to Systems of First order ODEs

● E.g. Harmonic oscillator:

● Introduce y=dx/dt:

● Resulting in a system of two first order ODEs● -> Sufficient to deal with systems of first order

ODEs!

d2

dt 2 x+k2 x=0

dx /dt= y

dy /dt=d2/dt 2 x=−kx2

Linear vs. Nonlinear

● A differential equation is linear if the unknown dependent variable and its derivatives only appear in a linear manner● Logistic growth and radioactive decay and harmonic

osci are linear equations● Pendulum and Lotka-Volterra are nonlinear

(Non)Autonmous Equations

● A differential equation is autonomous it does not explicitely depend on time● E.g.:

● All given examples in this lecture (Malthusian and logistic growth, harmonic osci, LV are autonomous)

● Non-autonomous equations can be transformed into systems of autonomous equations by including time as an independent variable

ddt

x=x autonomous

ddt

x=xt (non-)autonomous

Nonautonomous -> autonomous

● For example:

is non-autonomous. Include y=t and write:

● Is a system of two autonomous equations!

ddt

x=xt

ddt

y=1

ddt

x=xy

Transformations

● These trivial transformations● Higher order -> system of first order ODEs● Non-autonomous -> system of autonomous ODEs

are important as often ● Theory is only given for systems of first order ODEs● (Numerical) solvers often assume systems of first

order autonomous ODEs

Partial Differential Equations

● Differential equations can have unknown functions of multiple variables and their derivatives

● E.g. Heat equation

● Often solved numerically: finite difference methods, etc.

∂

∂ tu−α ∇

2u=0

uxx+u yy+uzz Temperature fieldThermal diffusivity

Summary

● You should remember:● What a differential equation is ● What solving a diff eq. means and how to verify if a

function is a solution to a diff. eq.● Classifications of differential equations and● Important tricks to transform diff. Eq's into systems

of first order ODEs