Differential Equations

Dillon & FadynSpring 2000

What is a differential equation?

An equation with derivatives in it.

Examples

ydx

dy2

)cos(5'3'' xyyy

r

u

rr

u

t

u

1

2

2

What was that last line?

That is a partial differential equation (P.D.E.) because it has partial derivatives.

r

u

rr

u

t

u

1

2

2

Partial Derivatives?

When u is a function of two variables, r and t, it has two partial first derivatives,

one with respect to r, one with respect to t.

Find the partial derivative of u with respect to t by holding r constant and differentiating as usual.

Example

22 )ln(),( trtrtru

Suppose

Then the partial derivative of u with respect to t would be

tt

rtr

rtr

t

u22

1 22

Notation

tut

u

ttut

u

t

u

t

2

2

trutr

u

t

u

r

2

Lingo

ttut

u

t

u

t

2

2

rttr urt

u

r

u

tu

tr

u

t

u

r

22

,

Second partial derivative of u with respect to t

Second mixed partial derivatives

Dealing with Differential Equations

• Determine what the dependent variable is when you are presented with a differential equation.

• Determine what the independent variable(s) is (are), too.

Example

ydx

dy2

Dependent variable

y

Independent variable

x

Why?

is a function ofy x

yis dependent onx

ydxdy 2/ In the equation

yis the dependent variable,

xis the independent variable.

Example

)cos(5'3'' xyyy

Dependent variable

y

Independent variable

x

Why?

are all functions of'',', yyy x

yis the dependent variable,

xis the independent variable.

In the equation )cos(5'3'' xyyy

as evidenced by the right hand side of the equation.

Example

r

u

rr

u

t

u

1

2

2

Dependent variable

u

Independent variables

tr,

Why?is a function ofu r and t

as evidenced by the partial derivatives

t

u

andr

u

uis the dependent variable,

tr,are the independent variables.

Why Does This Matter?

• We want solutions to differential equations.

• A solution to a differential equation is a function of the independent variable(s) which can successfully play the role of the dependent variable in the differential equation.

In Other Words

The unknown in a differential equation is the dependent variable.

It is the thing we want to find.It is the thing whose derivatives

appear in the differential equation.It is a function expressed in terms of

the independent variable(s).

Example

xe2is a solution to y

dx

dy2

because

)(2)( 2

2x

x

edx

ed

Example

xx exe is a solution to 0'2'' yyy

because

0)(

2)(

2

2

xx

xxxx

exedx

exed

dx

exed

Check that this is true by calculating the derivatives!

Notice

In the last example 0'2'' yyy

yis the dependent variable,

but we can call the independent variable ,, tx

or anything we want.

Exercise

Rewrite 0'2'' yyy

so that the independent variable is clearly.t

Example

22ln yx is a solution to 0 yyxx uubecause

0)(ln)(ln

2

222

2

222

y

yx

x

yx

Check that for homework!

Homework

• Read Sections 1.1 and 1.2 in the text• Answer the question, ``What is in the

text that we didn’t cover in class?’’• On page 8, do problems 1-5, 10, 12,

13, 16-21• On page 15, do problems 1, 2, 5, 11,

13, 16, 18-22, 28-30, 35, 39