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Chapter 4: Higher-Order Differential Equations

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Chapter 4: Higher-Order Differential Equations. Chapter 4: Higher-Order Differential Equations. 1. Sec 4.1: Linear DE (Basic Theory). Sec 4.1.1: Initial Value Problem (IVP) Boundary Value Problem (BVP). IVP:. . nth order linear DE. Theroem 4.1 ( Existence of a Unique Solution ).

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Presentation Transcript
slide3
1

Sec 4.1: Linear DE (Basic Theory)

Sec 4.1.1: Initial Value Problem (IVP)

Boundary Value Problem (BVP)

IVP:

. nth order linear DE

Theroem 4.1 ( Existence of a Unique Solution)

Sol y(x)

Exist

unique

slide4
1

Sec 4.1: Linear DE (Basic Theory)

Theroem 4.1 ( Existence of a Unique Solution)

Sol y(x)

Exist

unique

2

3

slide5
Sec 4.1: Linear DE (Basic Theory)

Theroem 4.1 ( Existence of a Unique Solution)

Sol y(x)

Exist

unique

Find an interval centered about x=0 for which the given IVP has a unique solution

9/p138

2

slide6
Sec 4.1: Linear DE (Basic Theory)

2ed order linear DE

Problem 1

Problem 2

What is the difference

IVP

BVP

slide7
Sec 4.1: Linear DE (Basic Theory)

2ed order linear DE

IVP

BVP

slide8
Sec 4.1: Linear DE (Basic Theory)

2ed order linear DE

IVP

BVP

Exist and unique

When??

slide9
BVP can have many, one, or No sol

BVP3

BVP2

BVP1

Given that

2-parameter family of solutions

unique

No sol

Infinity number of sol

slide10
Sec 4.1.2: Homogeneous Equations

diff

homogeneous

nonhomogeneous

1

(**) is the associated homogeneous DE of (*)

2

Remark: before we solve (*), we have to solve first (**)

slide11
Differential Operator

Differential Operators

slide13
Quiz on Monday

2.1

3.1

4.1.1

slide15
Homog DE

Theroem 4.2 ( Superposition Principle)

1)Constant multiple is sol

2)Sum of two sol is also sol

3) Trivial sol is also a sol ??

are solutions

slide16
Homog DE

In general

Theroem 4.2 ( Superposition Principle)

slide17
Linear Dependence & Linear Independence

Definition 4.1

IF

for every x in I

IF not then we say linearly independent

Note:Linear Combination

Is this set linearly dependent ??

slide18
Linear Dependence & Linear Independence

Definition 4.1

IF

for every x in I

IF not then we say linearly independent

Is this set linearly dependent ??

slide19
Linear Dependence & Linear Independence

Definition 4.1

IF

for every x in I

IF not then we say linearly independent

Special case

If a set of two functions is lin. Dep, then one function is simply a constant multiple of the other.

Is this set linearly dependent ??

slide20
Linear Dependence & Linear Independence

Definition 4.1

IF

for every x in I

IF not then we say linearly independent

Is this set linearly dependent ??

slide21
Linear Dependence & Linear Independence

Definition 4.1

IF

for every x in I

IF not then we say linearly independent

Remark

A set of functions is linearly dependent if at least one function can be expressed as a linear combination of the remaining

Is this set linearly dependent ??

slide22
Homogeneous Equations

We are interested to find n linearly independent solutions

of the homog DE

homogeneous

slide23
Wronskian

Definition 4.2

called the Wronskian of the functions

Compute the Wroskian of these functions

Compute the Wroskian of these functions

slide24
Criterion for Linearly Independent Solutions

Theroem 4.3

Linearly Independent

These functions are solutions for the DE

lin. Indep ?

slide25
Fundamental set of solutions

Def 4.3

Fundamental set of solutions

These functions are solutions for the DE

Fund. Set of sol. ?

These functions are solutions for the DE

Fund. Set of sol. ?

slide26
General Solution for Homog. DE

Theorem 4.5

Is the general solution for the DE.

These functions are solutions for the DE

Find the general sol?

general sol means what??

Given is a sol for

slide27
What is missing

How to solve Homog. DE

Given a homg DE:

Step 1

Find n-lin. Indep solutions

Step 2

The general solution for the DE is

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