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CHAPTER 5 Higher-Order Linear Differntial Equations

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CHAPTER 5 Higher-Order Linear Differntial Equations

Second-order DE:

(all)

Second-order linear DE:

(a,c,d,e)

Note: A,B,C,F function of x only

Second-order homogeneous linear DE:

Note: F(x)=0

(d,e)

Second-order homogeneous linear DE (with constant coefficients):

(d)

Note: A, B, C, are constants

CHAPTER 5 Higher-Order Linear Differntial Equations

Second-order homogeneous linear DE:

Note: F(x)=0

(d,e)

are solutions ?? (verify)

Consider the homogeneous 2ed-order linear DE:

(*)

Let W = the set of all solutions of (*)

Let F = the set of all real-valued functions

W is a subspace of F

dim(W)=2

CHAPTER 5 Higher-Order Linear Differntial Equations

Second-order homogeneous linear DE:

Consider the homogeneous 2ed-order linear DE:

(*)

W is a subspace of F

dim(W)=2

Give me other solutions???

CHAPTER 5 Higher-Order Linear Differntial Equations

How to solve homog. 2ed-order linear DE:

Consider the homogeneous 2ed-order linear DE:

(*)

1

Find two linearly independent solutions for (*)

2

The general solution for (*)

CHAPTER 5 Higher-Order Linear Differntial Equations

How to y1 & y2:

Consider the homogeneous

2ed-order linear DE (with constant coeff):

(*)

1

Find the characteristic equation

(**)

2

Find the roots of (**)

Distinct real

repeated real

2 non-real

CHAPTER 5 Higher-Order Linear Differntial Equations

How to y1 & y2:

Consider the homogeneous

2ed-order linear DE (with constant coeff):

(*)

1

Find the characteristic equation

(**)

2

Find the roots of (**)

Distinct real

repeated real

2 non-real

CHAPTER 5 Higher-Order Linear Differntial Equations

How to y1 & y2:

Consider the homogeneous

2ed-order linear DE (with constant coeff):

(*)

1

Find the characteristic equation

(**)

2

Find the roots of (**)

Distinct real

repeated real

2 non-real

CHAPTER 5 Higher-Order Linear Differntial Equations

How to solve homog. nth-order linear DE:

Consider the homogeneous nth-order linear DE:

(*)

1

Find n linearly independent solutions for (*)

2

The general solution for (*)

How to y1, y2, .. yn:

Consider the homogeneous

nth-order linear DE (with constant coeff):

1

Find the characteristic equation

(**)

2

Find the roots of (**)

Distinct real

repeated real

non-real

How to y1, y2, .. yn:

Consider the homogeneous

nth-order linear DE (with constant coeff):

1

Find the characteristic equation

(**)

2

Find the roots of (**)

Distinct real

repeated real

non-real

CHAPTER 5 Higher-Order Linear Differntial Equations

How to y1 & y2:

Consider the homogeneous

2ed-order linear DE (with constant coeff):

(*)

1

Find the characteristic equation

(**)

2

Find the roots of (**)

Distinct real

repeated real

2 non-real

How to y1, y2, .. yn:

Consider the homogeneous

nth-order linear DE (with constant coeff):

1

Find the characteristic equation

(**)

2

Find the roots of (**)

Distinct real

repeated real

non-real

CHAPTER 5 Higher-Order Linear Differntial Equations

Ploynomial Operator

Write in operator form

Euler’s Formula

CHAPTER 5 Higher-Order Linear Differntial Equations

How to solve homog. nth-order linear DE:

Consider the homogeneous nth-order linear DE:

(*)

Find n linearly independent solutions for (*)

1

The general solution for (*)

2

How to solve non-homog. nth-order linear DE:

Consider the non-homogeneous nth-order linear DE:

(**)

Solve the associated homog. DE (*)

1

(complementary function)

2

Find a particular solution for (**)

3

The general solution for (**)

CHAPTER 5 Higher-Order Linear Differntial Equations

How to solve homog. nth-order linear DE:

Consider the non-homogeneous nth-order linear DE:

(**)

Solve the associated homog. DE (*)

1

(complementary function)

2

Find a particular solution for (**)

3

The general solution for (**)