Unit Seven: Differential Equations

Basic steps in creating a model

1. State assumptions

2. Completely describe the variables and parameters used

3. Use assumptions to derive equations relating quantities

1st Order Equation

An equation for an unknown function in terms of its derivative: dy/dt=f(t,y)

Only contains 1st derivative of dependent variable.

Ordinary differential equation

Does not contain partial derivatives

Equilibrium solution

dP/dt=0

Unlimited population growth model

dP/dt=kP

Limited resources population growth model

dP/dt=k(1-P/N)P

N is the maximum population

A solution to a differential equation

y(t) is a solution if it satisfies

dy/dt=y'(t)=f(t,y(t))

An initial value problem

A differential equation along with an initial condition

dy/dt=f(t,y), y(t₀)=y₀

Separable equations

An equation is separable if the function f(t,y) can be written

dy/dt=f(t,y)=g(t)h(y)

An autonomous equation (and slope field)

dy/dt=f(t,y)=h(y)

The slope is the same at any t coordinate

Solving separable equations

dy/dt=f(t,y)=g(t)h(y)

1/h(y)dy=g(t)dt

∫1/h(y)dy=∫g(t)dt

Existence theorem (technical)

Suppose f(t,y) is a continuous function in a rectangle of the form {(t,y)|a

Existence theorem (not technical)

If f(t,y) is reasonable, then solutions exist. It does not rule out the possibility that solutions exist even if f(t,y) isn't nice. The solution may be defined for only a short interval of time

Uniqueness theorem (technical)

Suppose f(t,y) and df/dy are continuous functions in a rectangle of the form {(t,y)|a

Existence and Uniqueness theorem

There exists only one solution to a 1st order differential equation which satisfies a given initial condition. By this theorem, a solution that approaches an equilibrium point never actually gets there. Solutions move slowly as they get close to an equilibrium point.

Linearisation theorem

If f'(y₀)

If f'(y₀)>0, y₀ is a source

If f'(y₀)=0 then we need additional information to determine the type of equilibria

Parameter

Parameters are quantities that do not depend on the independent variable but that assume different values depending on the specifics of the application at hand.

A bifurcation

Is when a small change in the parameter can lead to a drastic change in the long-term solutions. We can say that a differential equation that depends on a parameter bifurcates if there is a qualitative change in the behaviour of solutions as the parameter changes.

Linear differential equations

Is a first order differential equation that can be written in the form:

dy/dt=a(t)y+b(t)

Where a(t) and b(t) are arbitrary functions of t

Homogenous/ unforced linear equations

b(t)=0

dy/dt=a(t)y

Constant coefficient linear differential equations

a(t) is constant

dy/dt=ay(t)+b(t)

Linearity Principle (Homogenous Case)

If y(t) is a solution of the homogenous linear equation,

dy/dt=a(t)y

Then any constant multiple of y(t) is also a solution

ky(t) is a solution for constant k

Extended Linearity Principle (Propositions)

Consider the non-homogenous equation dy/dt=a(t)y+b(t) and its associated homogeneous equation dy/dt=a(t)y.

1) If yh(t) is any solution of the homogenous equation and yp(t) is any solution of the non-homogenous equation, then yh(t)+y(p)t is also a solution of the non-homogenous equation

2) Suppose yp(t) and yq(t) are two solutions of the non-homogenous equation. Then yp(t)-yq(t) is a solution of the associated homogenous equation

Extended Linearity Principle (Conclusions)

If yh(t) is non-zero, kyh(t)+yp(t) is the general solution to the non-homogeneous equation.

Solving Linear Differential Equations (Integrating Factor Method)

dy/dt+g(t)y=b(t)

Integrating factor = µ(t)=exp(∫g(t)dt)

µ(t)dy/dt+µ(t)g(t)y=µ(t)b(t)

dy/dt(y.µ(t))=b(t)µ(t)

y(t)=1/(µ(t))∫µ(t)b(t)dt

Solving Linear Differential Equations (Linearity Principle Method)

1) Find the general solution to the associated homogeneous DE

2) Find one particular solution to the non-homogeneous DE (guess)

3) Obtain one general solution to the non-homogeneous DE by adding the general solution to the homogeneous DE and the particular solution the the non-homogeneous DE.

Limited resources population model with harvesting

dP/dt=kP(1-P/N)-H

H is harvested

Euler's Method

dy/dt=f(t,y), y(t₀)=y₀

t₁=t₀+∆t

y₁=y₀+f(t₀,y₀)∆t

Improved Euler Method

(a) take an ordinary Euler step of length h. Calculate the slope at the end of this step

(b) go back and take a step of length h with the slope being the average of the slope at the beginning and the slope at the end

4th Order Runge-Kutta

tⁿ⁺¹=tⁿ+h

m₁=f(tⁿ+(h/2),yⁿ+(h/2)m₁)

m₂=f(tⁿ+(h/2),yⁿ+(h/2)m₂)

m₃=f(tⁿ+(h/2),yⁿ+(h/2)m₂

m₄=f(tⁿ+h,yⁿ+hm₃)

yⁿ⁺¹=yⁿ+h/6(m₁+2m₂+2m₃+m₄)

Order of Numerical Methods

Measures the change in error of numerical solution as step size is decreased. Define E(h) to be the error in the approximate solution obtained when we solve the IVP

|E(h)|≈khⁿ as h→0. n is the order of the method.

Order of Various Numerical Methods

Euler: 1 : E≈kh¹ - error is halved when step size is halved

Improved Euler: 2: E≈kh²

R-K4: 4: E≈kh⁴

Solution to System of Differential Equations

x(t), y(t) is a solution if

dx/dt=x'(t)=f(t,x(t))

dy/dt=y'(t)=f(t,y(t))

Vector Notation

dx/dt=f(x,y) dy/dt=g(x,y)

Y(t)=(x(t) y(t))

V(Y)=(f(x,y) g(x,y))

dY/dt=V(Y)

The Vector Field

V(Y) is known as a vector field, i.e. the function that assigns a vector to each point in the x-y plane

Drawing vector fields for 2D autonomous systems

1) for selected points in the x-y plane, calculate V(Y)

2) For each point Y₀(x₀) draw the vector V(Y₀) with the base of the vector at Y₀ and with the arrow showing the direction. The length is proportional to size

Direction Field

Vectors have the same direction as in the vector field (x-y plane), but are scaled to a uniform length and arrows are often omitted

Solution curves as functions of t

We could plot solutions of a system of equations as functions of t. For a system of two dependent variables, x and y, we could either plot x(t) and y(t) separately, or we could plot the solution in the 3D x-y-t space

Phase Space

The space of dependent variables. In this case it is the x-y plane. It is the preferred method to plot solutions for an autonomous system

Sketching Solution Curves

dV/dt=F(Y), Y=(x(t) y(t))

A solution is a vector of functions

γ(t)=(y₁(t) y₂(t)) such that dγ/dt=F(γ)

As we vary t, the point (y₁(t) y₂(t)) will trace out a curve in the x-y plane. this curve is parameterised by time and is called the solution curve.

You can sketch solution curves for a DE by

1) Plot the direction field

2) Starting at some initial point, sketch a smooth curve that follows the line segments of the direction field.

Equilibrium solutions of DEs

Y₀ is an equilibrium point of the system dY/dt=V(Y) if V(Y₀)=0.

The intersection of the x and y nullclines is an equilibrium

X Nullcline

Consider a system dx/dt=f(x,y), dy/dt=g(x,y)

The x-nullcline is the set of all points (x,y) where f(x,y)=0. On the x-nullcline, dx/dt=0 and the direction field is vertical (pointing straight up or down)

Y Nullcline

Consider a system dx/dt=f(x,y), dy/dt=g(x,y)

The y-nullcline is the set of all points (x,y) where g(x,y)=0. On the y-nullcline, dy/dt=0 and the direction field is horizontal (pointing left or right)

Points to note about nullclines

1) Nullclines are not necessarily a solution curve

2) Nullclines are not necessarily a straight line

When is f(t,y)=y√t continuous?

For all y∈R and for all t>0