Diff Eqns

Differential Equations

Brannan
Copyright 2010 by John Wiley & Sons, Inc.

All rights reserved.

Chapter 01: Introduction

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Chapter 1 Introduction

Chapter 1
Introduction

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1.1 Mathematical Models, Solutions, and Direction Fields

Differential Equations Real world relations involving rates at which one variable, say y, may change with respect to another variable, t. Most often, these relations take the form of equations containing y and certain of the derivatives y', y'' , . . . , y(n) of y with respect to t. The resulting equations are then referred to as differential equations.

Mathematical Models

Ex: Heat Transfer: Newtons Law of Cooling.

du/dt= k(u T ), or u' = k(u T ).

Terminology

- time t is an independent variable,

- temperature u is a dependent variable because it depends on t,

- k and T0 are parameters in the model.

Terminology (Ctd.)

Several integral curves of
u' = 1.5(u 60).

The general solution to

This diff. eq. is

u = 60 + ce1.5t .

c=10 is the solution satisfying initial condition u(0)=70.

The Direction Field for
y' = f(t,y)

A solution of dy/dt = f (t, y) is a differentiable function y = (t) such that '(t) = f (t, (t)).

Hence f (t, y) specifies the slope of a solution passing through the point y at time t. This information can be displayed geometrically by drawing a line segment with slope f (t, y) at the point (t, y).

A direction field (or Slope filed) consists of a large number of such line segments, usually drawn on a rectangular grid. A direction field provides a geometric representation of the flow of solutions of the differential equation.

Direction field and equilibrium solution u = 60 for u' = 1.5(u 60).

A solution that does not change with time is called the equilibrium solution.

Example - Heating and Cooling of a Building

Equation: du/dt + ku = kT0 + kA sin t

Solution: u = T0 + kA/(k2 + 2) (k sin t cos t) + cekt

Example: Let k = 1.5 day1, T0 = 60F, A = 15F, and = 2 be given.

Then, du/dt + 1.5u = 90 + 22.5 sin 2t (25)

with corresponding general solution

u = 60 + 22.5/(2.25 + 42) (1.5 sin 2t 2 cos 2t) + ce1.5t .

Example - Heating and Cooling of a Building (Ctd)

The steady-state solution

is the solution to the equation

As t . In this example,

it is:

U(t) = 60 + (22.5/2.25 + 42)

(1.5 sin 2t 2 cos 2t)

1.2 Linear Equations: Method
of Integrating Factors

DEFINITIONS:

A differential equation that can be written in the form (standard form)

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

is said to be a first order linear equation in the dependent variable y.

If h(t)=0, it is said to be homogeneous; otherwise, the equation is nonhomogenous. If powers of y are there equation becomes nonlinear.

The Method of Integrating Factors for Solving y' + p(t)y = g(t).

1. Put the equation in standard form:

y' + p(t)y = g(t).

2. Calculate the integrating factor

(t) = ep(t)dt

3. Multiply the equation by (t) and write it in the form [(t)y]' = (t)g(t).

4. Integrate this equation to obtain

(t)y = (t)g(t) dt + c.

5. Solve for y.

Example

Solve the initial value problem

ty' + 2y = 4t2, y(1) = 2.

1. Standard form: y' + (2/t)y = 4t.

2. Integrating factor (t) = e2/tdt =t2

3. Multiply the equation by (t) and write it in the form [t2y]' = 4t3

4. Integrate and obtain

t2y = t4+ c.

5. Solve for y. y = t2 + 1/t2 , t > 0

Integral Curves for the initial value problem ty' + 2y = 4t2

1.3 Numerical Approximations:
Eulers Method

When drawing direction field, many tangent line segments at successive values of t almost touch each other. By linking these an approximation to a solution of the differential equation can be obtained.

1. Can we carry out this linking of tangent lines in a simple and systematic manner?

2. If so, does the resulting piecewise linear function provide an approximation to an actual solution of the differential equation?

3. If so, can we say anything about the accuracy of the approximation?

It turns out that the answer to each question is affirmative.

Euler Formula

For the initial value problem

y = f (t, y), y(t0) = y0,

the Euler method is the numerical approximation algorithm

yn+1 = yn + f (tn, yn)(tn+1 tn),

n = 0, 1, 2, . . .

If uniform step size h is used.

tn+1 = tn + h, yn+1 = yn + h f (tn, yn),

To use Eulers method, you simply evaluate the equations repeatedly.

Euler Formula (Ctd.)

The result at each step is used to execute the next step. Thereby generate a sequence of values y1, y2, y3, . . . that approximate the values of the solution (t) at the points t1, t2, t3, . . . .

Instead of a sequence of points, to get a function to approximate the solution (t), we use the piecewise linear function constructed from the collection of tangent line segments.

That is, let y be given by y = y0 + f (t0, y0)(t t0) in [t0, t1], and in general y is given by

y = yn + f (tn, yn)(t tn) in [tn, tn+1].

Example: Euler Method

Problem: Use Eulers method with a step size h = 0.2 to approximate the solution of the initial value problem

dy/dt + 1/2y = 3/2 t, y(0) = 1 on the interval 0 t 1.

Answer: To use Eulers method, note that f (t, y) = 1/2y + 3/2 t, so using the initial values t0 =0, y0 = 1, we have

f0 = f (t0, y0) = f (0, 1) = 0.5 + 1.5 0 = 1.0.

Example: Euler Method (Ctd.)

The tangent line approximation for t in [0, 0.2] is y = 1 + 1.0(t 0) = 1 + t.

Setting t = 0.2, we find the approximate value y1 of the solution at t = 0.2, namely, y1 = 1.2.

At the next step we have

f1 = f (t1, y1) = f (0.2, 1.2) = 0.6 + 1.5 0.2 = 0.7, and y = 1.2 + 0.7(t 0.2) = 1.06 + 0.7t for t in [0.2, 0.4]. Evaluating at t = 0.4, we obtain y2 = 1.34.

Example: Euler Method (Ctd.)

1.4 Classification of Differential
Equations:
1. Ordinary and Partial Differential Equations

- If only ordinary derivatives appear in the differential equation then it is an ordinary differential equation.

- If derivatives are partial derivatives then the equation is called a partial differential equation.

2.Systems of Differential Equations

For example, the LotkaVolterra, or predatorprey, equations are important in ecological modeling. They have the form

dx/dt = ax xy

dy/dt = cy + xy

3. Order

ay'' + by' + cy = f (t),

where a, b, and c are given constants, and f is a given function, is a second order equation.

4. Linear and Nonlinear Equations

The ordinary differential equation F(t, y, y', . . . , y(n)) = 0 is said to be linear if F is a linear function of the variables y, y', . . . , y(n).Thus the general linear ordinary differential equation of order n is

a0(t)y(n)+ a1(t)y(n-1)+ +an(t)y = g(t).

An equation that is not of this form is a nonlinear equation.

Some Terminology

(n)(t) = f [t, (t), '(t), . . . , (n-1)(t)]

for every t in < t < .

Some Important Questions

Computer Use in Differential Equations

CHAPTER SUMMARY

Differential equations are used to model systems that change continuously in time. The mathematical investigation of a differential equation usually involves one or more of three general approaches: geometrical, analytical, and numerical.

1.1 Geometrical Approach

A solution of dy/dt = f (t, y) is a differentiable function y = (t) such that '(t) = f (t, (t)).

Hence f (t, y) specifies the slope of a solution passing through the point y at time t. This information can be displayed geometrically by drawing a line segment with slope f (t, y) at the point (t, y).

A direction field (or Slope filed) consists of a large number of such line segments, usually drawn on a rectangular grid. A direction field provides a geometric representation of the flow of solutions of the differential equation.

1.2 Linear Equations Analytical Approach

dy/dt + p(t)y = g(t).Multiplying by the integrating factor (t)=exp[p(t)dt] leads to d(y)/dt=g. Integrating then yields y = 1[g dt + c].

For the initial value problem y = f (t, y), y(t0) = y0, the Euler method is the numerical approximation algorithm tn+1 = tn + h, yn+1 = yn + h f (tn, yn), n = 0, 1, 2, . . . that geometrically consists of a finite number of connected line segments, each of whose slopes is determined by the slope at the initial point of each segment.

1.3 Numerical Approach

1.4 Classification of Differential Equations

Differential equations are classified according to various criteria for a sensible treatment of theory, solution methods, and solution behavior: