Lecture (25): Ordinary Differential Equations (1 of 2)

A differential equation is an algebraic equation that contains some derivatives:

037352

2

ydx

dy

dx

ydty

dt

dy

• Recall that a derivative indicates a change in a dependent variable with respect to an independent variable.

• In these two examples, y is the dependent variable and t and x are the independent variables, respectively.

Why study differential equations?

• Many descriptions of natural phenomena are relationships (equations) involving the rates at which things happen (derivatives).

• Equations containing derivatives are called differential equations.

• Ergo, to investigate problems in many fields of science and technology, we need to know something about differential equations.

Why study differential equations?

• Some examples of fields using differentialequations in their analysis include:

— Solid mechanics & motion

— heat transfer & energy balances

— vibrational dynamics & seismology

— aerodynamics & fluid dynamics

— electronics & circuit design

— population dynamics & biological systems

— climatology and environmental analysis

— options trading & economics

Examples of Fields Using Differential Equations in Their Analysis

CoolBath

HotBath

kydt

dyckx

dt

dxc

dt

xdm

2

2

02

2

kxdt

dxc

dt

xdm

ikkxdt

dxc

dt

xdm f

2

2

ghACdt

dhA od 2

70 Hkdt

dH

Differential Equation Basics

• The order of the highest derivative ina differential equation indicates the order of the equation.

D.E. PartialOrder Second 01

EquationOrder Second 037

EquationOrder First 35

2

2

2

2

t

T

x

T

ydx

dy

dx

yd

tydt

dy

Simple Differential Equations

A simple differential equation has the form

( )dy

f xdx

Its general solution is

( )y f x dx

Ex. Find the general solution to

Simple Differential Equations

dxxxdy 32 42

32 42 xxdx

dy

dxxxy 32 42

Cxxy 43

3

2

Ex. Find the general solution to

Simple Differential Equations

dxxx

dy

2

1

xxdx

dy2

1

dxx

xy 2

1

Cxxy 2ln

Find the general solution to

Exercise: (Waner, Problem #1, Section 7.6)

dxxxdy 2

xxdx

dy 2

dxxxdxxxy 2122 /

Cxxy 233

3

2

3

1 /

A drag racer accelerates from a stop so that its speed is 40t feet per second t seconds after starting. How far will the car go in 8 seconds?

Example: Motion

seconds.in timeis and

feet,in distance theis )( wher , 40

t

tstdt

ds

ft ?8 s

Given:

Find:

dttds 40

tdt

ds40

Ctdttts 22040

ft 12808208 2 s

Solution:

Apply the initial condition: s(0) = 0

Cs 202000 0C

220tts

The car travels 1280 feet in 8 seconds

Find the particular solution to

dxxxdy 23

0 when 1 ;23 xyxxdx

dy

Cxxdxxxy 243

4

12

14

1 24 xxy

Exercise: (Waner, Problem #11, Section 7.6)

Apply the initial condition: y(0) = 1

C 24 004

11 1C

Separable Differential Equations

A separable differential equation has the form ( ) ( )dy

f x g ydx

Its general solution is1

( )( )dy f x dx

g y

Consider the differential equation 2y

x

dx

dy

Example: Separable Differential Equation

a. Find the general solution.

b. Find the particular solution that satisfies the initial condition y(0) = 2.

dxxdyy 2

Solution:

Step 1 — Separate the variables:

dxxdyy2Step 2 — Integrate both sides:

Step 3 — Solve for the dependent variable:

Cxy

23

23

a.

312

312

2

33

2

3//

DxCxy

This is the general solution

Solution: (continued)

Apply the initial (or boundary) condition, that is, substituting 0 for x and 2 for y into the general solution in this case, we get

3131

202

32 /

/

DD

Thus, the particular solution we are looking for is

823 D

b.

312 8

2

3/

xy

Find the general solution to

Exercise: (Waner, Problem #4, Section 7.6)

x

dx

y

dy

x

y

dx

dy

x

dx

y

dy

) (where CeAAxy

Cxy lnln

xeey CCx ln

Axxey C