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
If has coefficient 1, we call this the standard form.
1.2 Homogemeous and nonhomogeneous
An ODE is called homogeneous if r(x) in (1) = 0. Then (1) become
If , 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 nonhomogeneous or nonlinear ODE!)
For example, consider an nth order ODEs with solutions
it also has a solution
1.4 Linear independence and dependence
The functions are called linearly independent on some interval I if the equation below holds on I
implies that all are zero. These functions are called linearly dependent on I if this equation holds
on I for some not all zero.
Example 1.4.
1.
2.
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
where 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 in (4) have specific values.
1.6 Initial value problem
An initial value problem for (2) consists of (2) and n initial conditions
For example,
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 , defined as the nth order determinant. The n solutions of (2) on I are linearly dependent on I if and only if their Wronskian
for some in I. Futhermore, if W = 0 for , then W is identically zero on I. Hence if there is an in I at which , then 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 is
So they are linearly independent on any interval.
Exercise 1.7. Are the given functions linearly dependent or independent on x > 0?
(More Exercises: Problem Set 3 question 1)
2 Homogeneous linear ODEs with constant coefficients
Consider nthorder homogeneous linear ODEs with constant coefficients,
Then substituting , we obtain the characteristic equation
of (7). If is a root of (8), is a solution of (7).
2.1 Distinct real roots
If all the n roots of (8) are real and different, then the n solutions
constitute a basis for all x. The corresponding general solution of (7) is
Example 2.1. Solve .
Solution. By substituting , we obtain the characteristic equation
Ex 2.1.1. Find the solutions of .
The roots are 1, 1, 2. Since they are different, the general solution is
Exercise 2.1. Solve .
2.2 Simple complex roots
If complex roots occur, we have two linearly independent solutions
Example 2.2. Solve the initial value problem
Exercise 2.2. Solve .
2.3 Multiple real roots
If multiple real roots occur  say we get m solutions from the characteristic equation , i.e., is a real root of order m, then m corresponding linearly independent solutions are
Example 2.3. Solve the ODE
Solution. The characteristic equation
has three roots .
Ex 2.3.1. Write down all the corresponding linearly independent solutions we have in this ODE.
The general solution should be
Exercise 2.3. Solve
2.4 Multiple complex roots
This is similar to the previous sections  if we have m complex roots , and also m conjugates , then the corresponding linearly independent solutions are
Example 2.4. Solve the .
Solution. Substituting , the characteristic equation is
The solutions are 1+i, 1+i, 1i and 1i. So, the general solution is
Exercise 2.4. Solve .
(More Exercises: Problem Set 3 question 2)
Summarize Table of section 2

Case 
Independent Solutions 
General Solution 
Distinct real roots 


Simple complex roots 

Dependent on questions 
Multiple real roots is a real
root of order m


Dependent on questions 
Multiple complex roots m complex root 

Dependent on questions 
3 Nonhomogeneous linear ODEs
Recall from (1), an nth order nonhomogeneous linear ODE is
Just like second order ODEs, the general solution of (10) on an open interval I of the xaxis is of the
form
is a general solution of the corresponding homogemeous ODE (see (2)). 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 form the corresponding homogeneous ODE.
Step 2. Find and get the general solution.
Step 3. If provided, use the initial conditions to find the particular solution.
To find , we use the same two methods as in the previous chapter (Sections 8.1, 8.2).
3.1 Method of undetermined coefficients
For a constantcoefficient equation
We can find 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 in (12) is one of the functions in the first column in the Table, choose in the same line and determine its undetermined coefficients by substituting and its derivatives into (12).
(b) Modification Rule. If a term in your choice for is a solution of the homogeneous equation , then multiply by , where is the smallest positive integer such that no term of is a solution of the homogeneous ODE.
(c) Sum Rule. If is a sum of function in the first column of the Table, choose for the sum of the functions in the corresponding lines of the second column.
Example 3.1. Solve
Solution.
Step 1. Find .
Ex 3.1.1. Find .
We get .
Step 2. Find
If we try , we get , which has no solution.
Ex 3.1.2. Try and see what we get.
The Modification rule calls for
By substituting its derivatives into the ODE, we have C=5. So
and the solution is
Exercise 3.1. Solve .
ADVANCED TOPICS
3.2 Method of variation of parameters
This method gives a particular solution for the nonhomogeneous equation (10) in standard form by the formula
on an open interval I on which the coefficients of (10) and are continuous.
Example 3.2. Solve the nonhomogeneous EulerCauchy equation
Solution.
Step 1. Find .
Substitution of and the derivatives into the homogeneous ODE and deletion of the factor gives
The roots are 1, 2, 3. The general solution of the homogeneous ODE is
Step 2. Find .
In this case, we use the method of variation of parameters.
First find W,
Ex 3.2.1. Find .
Then write the ODE in standard form and we get .
By the formula (13),
Hence the answer is
Exercise 3.2. Solve
(More Exercises: Problem Set 3 question 3, 4)
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