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Chapter 3: Higher order linear ODEs

Page history last edited by sidjaggi 13 years, 5 months ago

Chapter 3: Higher order linear ODEs

 

In this chapter we extend the concepts and methods of chapter 2 for linear ODEs from order n = 2 to arbitrary order n.

 

1 Basic concepts

 

1.1 Linear

 

An ODE is called linear if it

Formula 

If Formula has coefficient 1, we call this the standard form.

 

1.2 Homogemeous and non-homogeneous

 

An ODE is called homogeneous if r(x) in (1) = 0. Then (1) become

Formula 

If Formula, the ODE is called nonhomogeneous.

 

1.3 Superposition principle

 

For a homogeneous linear ODE (2), sums and constant multiples of solution on some open interval I are again solutions on I. (This does not hold for a non-homogeneous or nonlinear ODE!)

     For example, consider an nth order ODEs with solutions

Formula

it also has a solution

Formula

 


1.4 Linear independence and dependence

 

The functions Formula are called linearly independent on some interval I if the equation below holds on I

Formula 

implies that all Formula are zero. These functions are called linearly dependent on I if this equation holds

on I for some Formula not all zero.

 

Example 1.4.

 

1. Formula

2. Formula

 

1.5 General solution, basis, particular solution

 

A general solution of (2) on open interval I is a solution of (2) on I of the form

Formula 

where Formula is a basis of solution of (2) on I; that is , these solutions are linearly independent on I.

A particular solution of (2) on I is obtained if the n constants Formula in (4) have specific values.

 

1.6 Initial value problem

 

An initial value problem for (2) consists of (2) and n initial conditions

Formula 

 

For example,

Formula

It is a third order ODE and 3 initial conditions.

 

1.7 Linear independence of solutions, Wronskian

 

This extended criterion uses the Wronskian W of n solutions Formula, defined as the nth order determinant. The n solutions Formula of (2) on I are linearly dependent on I if and only if their Wronskian

Formula

 

for some Formula in I. Futhermore, if W = 0 for Formula, then W is identically zero on I. Hence if there is an Formula in I at which Formula, then Formula are linearly independent on I, and they form a basis of solutions of (2) on I.

Note: When we take n = 2, it is the same as in Chapter 2.

 

Example 1.7. Consider the Example 2 in Section 1.4, the Wronskian of Formula is 

Formula

So they are linearly independent on any interval.

 

Exercise 1.7. Are the given functions linearly dependent or independent on x > 0?

Formula

 

 

(More Exercises: Problem Set 3 question 1)

 

2 Homogeneous linear ODEs with constant coefficients

 

Consider nth-order homogeneous linear ODEs with constant coefficients,

Formula 

Then substituting Formula, we obtain the characteristic equation 

Formula 

of (7). If is a root of (8), Formula is a solution of (7). 

 

2.1 Distinct real roots

 

If all the n roots Formula of (8) are real and different, then the n solutions

Formula

constitute a basis for all x. The corresponding general solution of (7) is

Formula 

 

Example 2.1. Solve Formula.

Solution. By substituting Formula, we obtain the characteristic equation

Formula 

 

Ex 2.1.1. Find the solutions of Formula.

 

 

The roots are -1, 1, 2. Since they are different, the general solution is

Formula 

 

Exercise 2.1. Solve Formula.

 

 

2.2 Simple complex roots

 

If complex roots Formula occur, we have two linearly independent solutions

Formula 

 

Example 2.2. Solve the initial value problem

Formula

 

Exercise 2.2. Solve Formula.

 

 

 

2.3 Multiple real roots

 

If multiple real roots occur -- say we get m solutions Formula from the characteristic equation Formula, i.e., Formula is a real root of order m, then m corresponding linearly independent solutions are

Formula 

 

Example 2.3. Solve the ODE Formula

Solution. The characteristic equation 

Formula 

has three roots Formula.

Ex 2.3.1. Write down all the corresponding linearly independent solutions we have in this ODE.

 

 

The general solution should be

Formula 

 

Exercise 2.3. Solve Formula

 

 

2.4 Multiple complex roots

 

This is similar to the previous sections -- if we have m Formula complex roots Formula, and also m conjugates Formula, then the corresponding linearly independent solutions are

Formula

 

Example 2.4. Solve the Formula.

Solution. Substituting Formula, the characteristic equation is

Formula

The solutions are 1+i, 1+i, 1-i and 1-i. So, the general solution is 

Formula 

 

Exercise 2.4. Solve Formula.

 

 

(More Exercises: Problem Set 3 question 2)

 

Summarize Table of section 2
Case  Independent Solutions  General Solution 
Distinct real roots  Formula  Formula 
Simple complex roots Formula

Formula 

Formula

Dependent on questions 

Multiple real roots Formula is a real

root of order m Formula

Formula 

Formula

Dependent on questions 
Multiple complex roots m Formula complex root Formula

Formula 

Formula

Formula

 

Dependent on questions 

 

3 Non-homogeneous linear ODEs

 

Recall from (1), an nth order nonhomogeneous linear ODE is

Formula 

Just like second order ODEs, the general solution of (10) on an open interval I of the x-axis is of the

form

Formula 

Formula is a general solution of the corresponding homogemeous ODE (see (2)). Formula is any solution of (10) on I containing no arbitrary constants.

     The steps to solve nth order nonhomogeneous ODEs are the same for second order ones.

 

Step 1. Find Formula form the corresponding homogeneous ODE.

Step 2. Find Formula and get the general solution.

Step 3. If provided, use the initial conditions to find the particular solution.

 

To find Formula, we use the same two methods as in the previous chapter (Sections 8.1, 8.2).

 

3.1 Method of undetermined coefficients

 

For a constant-coefficient equation 

Formula 

We can find Formula by the method of undetermined coefficients.

     To use this method, we need to follow three rules. It is very simiplar to section 8.1.1 in chapter 2. The only difference is the Modication rule.

 

(a) Basic Rule.     If Formula in (12) is one of the functions in the first column in the Table, choose Formula in the same line and determine its undetermined coefficients by substituting Formula and its derivatives into (12).

 

(b) Modification Rule.     If a term in your choice for Formula is a solution of the homogeneous equation Formula, then multiply Formula by Formula, where Formula is the smallest positive integer such that no term of Formula is a solution of the homogeneous ODE.

 

(c) Sum Rule.     If Formula is a sum of function in the first column of the Table, choose for Formula the sum of the functions in the corresponding lines of the second column.

 

Formula

 

Example 3.1. Solve Formula

Solution.

Step 1. Find Formula.

     Ex 3.1.1. Find Formula.

 

 

     We get Formula.

Step 2. Find Formula

     If we try Formula, we get Formula, which has no solution.

     Ex 3.1.2. Try Formula and see what we get.

 

 

     The Modification rule calls for     

 Formula

     By substituting its derivatives into the ODE, we have C=5. So      

Formula 

     and the solution is

 

Formula 

 

Exercise 3.1. Solve Formula.

 

 

ADVANCED TOPICS

3.2 Method of variation of parameters

 

This method gives a particular solution Formula for the nonhomogeneous equation (10) in standard form by the formula 

Formula 

on an open interval I on which the coefficients of (10) and Formula are continuous. 

 

Formula

 

Example 3.2. Solve the nonhomogeneous Euler-Cauchy equation

Formula 

Solution.

     Step 1. Find Formula

     Substitution of Formula and the derivatives into the homogeneous ODE and deletion of the factorFormula gives     

Formula 

 The roots are 1, 2, 3. The general solution of the homogeneous ODE is      

Formula 

     Step 2. Find Formula.

     In this case, we use the method of variation of parameters.

     First find W,

     Formula

 

 

     Ex 3.2.1. Find Formula.

 

 

     Then write the ODE in standard form and we get Formula.

     By the formula (13), 

     Formula

     Hence the answer is

     Formula

 

Exercise 3.2. Solve Formula

 

 

(More Exercises: Problem Set 3 question 3, 4)

 

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