# MAT244H1- Midterm Exam Guide - Comprehensive Notes for the exam ( 34 pages long!)

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UTSG

MAT244H1

MIDTERM EXAM

STUDY GUIDE

MAT244H1a.doc

Page 1 of 32

Introduction

I

NTRODUCTION

T

O

D

IFFERENTIAL

E

QUATIONS

A differential equation is an equation involving some hypothetical function and its derivatives.

Example

xyy =

′

+

′

′

2

is an differential equation. As such, the differential equation is a description of some function

(exists or not).

A solution to a differential equation is a function that satisfies the differential equation.

Example

xxy 5

3

+=

is a solution to

3

xyxy −+=

′′

.

Some differential equations are famous/important:

•

yy =

′

,

x

ey =

.

•

ayy =

′

,

ax

ey =

.

•

0=+

′

′

yy

,

x

y

cos

=

.

•

0=+

′

′

ayy

,

xay cos=

.

Recall that a differential equation describes a phenomenon in terms of changes. For example, if

mvP

=

, then

vdmFdt ⋅=

or

F

dt

dP =

.

Example

A pool contains V liters of water which contains M kg of salt. Pure water enters the pool at a constant rate of v

liters per minute, and after mixing, exits at the same rate. Write a differential equation that describes the

density of salt in the pool at an arbitrary time t.

• Let

(

)

t

ρ

be the density at time t. Then

( )

(

)

V

tM

t=

ρ

.

• To model change in

(

)

t

ρ

, let

(

)

11

t

ρρ

=

and

(

)

22

t

ρρ

=

. Then

(

)

(

)

12112

ttvV −−≈−

ρρρ

, so

V

v

V

tt

1

12

12

ρ

ρρ

−≈

−

−

or

(

)

(

)

( )

V

v

tV

ttt

ttt

ρ

ρρ

−≈

−∆+

−∆+

.

• Now, as

0

→

t

,

( )

V

v

t

dt

d

ρ

ρ

−=

or

V

v

ρρ

−=

′

.

• Without solving this equation, we can predict facts about this system. As

∞

→

t

,

0

→

ρ

.

•

To solve this differential equation, write

dt

V

v

d−=

ρ

ρ

. Integrating both sides, we get

( ) ( )

t

V

v

t

V

v

C

Ct

V

v

AeeeetCt

V

v

t

−−+−

===+−=

ρρ

ln

.

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MAT244H1a.doc

Page 2 of 32

•

If we add

(

)

ρρ

=0

to

V

v

ρρ

−=

′

, then we have an IVP (initial value problem).

I

SSUES

A

BOUT THE

U

SE OF

D

IFFERENTIAL

E

QUATIONS

1)

How to translate a real problem to a differential equation. Keep your eyes open!

2)

Some patterns of nature are ill-defined. Use different points of view and different mental differential equation

models to reformulate them.

3)

Differential equations have infinitely many solutions. Which one is yours? The initial value are extremely

important.

4)

There may be no analytic solution found.

•

Is there a solution?

•

Is this solution unique?

•

If a numeric answer is required, i.e. the value of the solution at one particular point, then use numerical

approximations. It does not give any feelings for the pattern, nor does it give elbow room.

•

Use theoretical analysis if you need the behavior of the solution. This does not give any values.

•

To know the behavior locally/in a neighborhood, solve in series.

5)

The data does not fit you solution. You need to repeat (as in a feedback/controlled system).

N

OTATIONS

W

ITH

R

EGARD TO THE

I

NPUT

/O

UTPUT

S

YSTEMS

Example

xyxyyx tansin2 =⋅−

′

+

′

′

can be written as

[

]

xyL tan=

. Solve it, and the answer is the output.

•

[

]

yL

is the “black box system”.

•

xtan

is the ‘input”.

For theoretical purposes, mathematicians use these equivalents:

x

y

y

x

x

x

x

y

′

−+=

′′′

2

sintan

is the same as

(

)

yyxfy

′

=

′

′

′

,,

or

(

)

0,,,,

=

′

′

′

′

′

′

yyyyxF

.

L

INEAR VS

.

N

ON

-L

INEAR

D

IFFERENTIAL

E

QUATIONS

•

x

yey

x

yx

x

tan1

11

tan

+

=+

′

+

′′

⋅

is linear.

•

0

2=+

′

+

′′′ yyy

is non-linear.

First Order Differential Equations

L

INEAR

E

QUATIONS

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## Document Summary

A differential equation is an equation involving some hypothetical function and its derivatives. As such, the differential equation is a description of some function (exists or not). A solution to a differential equation is a function that satisfies the differential equation. 3 + x y x is a solution to y y x. Some differential equations are famous/important: y = y = xe y axe y = y = ay. 0=+ y y y cos x y ay. Recall that a differential equation describes a phenomenon in terms of changes. A pool contains v liters of water which contains m kg of salt. Pure water enters the pool at a constant rate of v liters per minute, and after mixing, exits at the same rate. Write a differential equation that describes the density of salt in the pool at an arbitrary time t. ( )t : let, to model change in v. 1 be the density at time t. then.