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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)

Sol y(x)

Exist

unique

1

Sec 4.1: Linear DE (Basic Theory)

Theroem 4.1 ( Existence of a Unique Solution)

Sol y(x)

Exist

unique

2

3

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

Sec 4.1: Linear DE (Basic Theory)

2ed order linear DE

Problem 1

Problem 2

What is the difference

IVP

BVP

Sec 4.1: Linear DE (Basic Theory)

2ed order linear DE

IVP

BVP

Sec 4.1: Linear DE (Basic Theory)

2ed order linear DE

IVP

BVP

Exist and unique

When??

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

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 (**)

Differential Operator

Differential Operators

Properties: Differential Operator

Linear Operator

Quiz on Monday

2.1

3.1

4.1.1

DE Differential Operator Form

Write as DE

where

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

Homog DE

In general

Theroem 4.2 ( Superposition Principle)

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 ??

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 ??

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 ??

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 ??

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 ??

Homogeneous Equations

We are interested to find n linearly independent solutions

of the homog DE

homogeneous

Wronskian

Definition 4.2

called the Wronskian of the functions

Compute the Wroskian of these functions

Compute the Wroskian of these functions

Criterion for Linearly Independent Solutions

Theroem 4.3

Linearly Independent

These functions are solutions for the DE

lin. Indep ?

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. ?

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

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