1. Chapter 4: Linear Differential Equations
MATH 374 Lecture 10
Chapter 4: Linear Differential Equations

2. 4.1: The General Linear Equation
Definition: An equation that can be written in the form
is called a general linear equation of order n.
If R(x) is zero for all x, (1) is said to be homogeneous.
If b0(x), b1(x), … , bn(x), and R(x) are continuous on interval I and b0(x) is never zero on I, we say (1) is normal on I.
Example 1: The equation
is a homogeneous, second order, linear differential equation.
On any interval not containing x = -2, it is normal.

3. Linear Combination of Functions
Definition: A linear combination of the functions {f1, f2, … , fk } is a function of the form:
c1 f1 + c2 f2 + … + ck fk
where c1, c2, … , ck are constants.
An important property of linear, homogeneous differential equations is the Principle of Superposition! 3

4. Principle of Superposition
Theorem 4.1: Any linear combination of solutions of a homogeneous linear differential equation is also a solution.
Proof:
We will look at the case when we have a linear combination of two solutions.
For more than two solutions, use induction.
(Assume true for n  2 to show true for n+1.) 4

5. Principle of Superposition
(1) With R(x) ´ 0
Proof of Theorem 4.1 (continued):
Assume y1 and y2 are solutions of (1) with R(x) ´ 0.
Then for any constants c1 and c2, letting
y = c1 y1 + c2 y2,
we see that: 5

6. Principle of Superposition
(1) With R(x) ´ 0
Proof of Theorem 4.1 (continued):  6

7. 4.2: An Existence and Uniqueness Theorem
We state without proof, an existence and uniqueness theorem for general linear equations (compare to Theorem 2.4 for the case when n = 1).
A proof for this theorem can be found in Coddington’s An Introduction to Ordinary Differential Equations – see Chapter 6 (beyond the scope of this course). 7

8. Existence and Uniqueness
Theorem 4.2: Given an nth order linear differential equation
that is normal on the interval I,
suppose that x0 is any number in I and y0, y1, … , yn-1 are arbitrary real numbers.
Then a unique function y = y(x) exists such that y is a solution of (1) on I and y satisties the initial conditions
y(x0) = y0, y’(x0) = y1, … , y(n-1)(x0) = yn-1. 8

9. Example 1 Given that y1 = x3 and y2 = x-3 are solutions of
x2 y’’ + x y’ – 9y = 0, (2)
find the unique solution to (2) with the initial data:
y(1) = -1; y’(1) = (3) 9

10. Example 1 (continued) 10 Solution:
x2 y’’ + x y’ – 9y = 0 (2)
y(1) = -1; y’(1) = 15 (3)
Example 1 (continued)
Solution:
By Theorem 4.2, a unique solution to (2), (3) exists on any interval containing x = 1, for which (2) is normal.
Therefore a unique solution to (2), (3) exists on (0, 1).
From Theorem 4.1, we know
y = c1 y1 + c2 y2 = c1 x3 + c2 x-3
must be a solution of (2), for any choice of constants c1 and c2.
Using this information and (3), we can find the unique solution to (2), (3)! 10

11. Example 1 (continued) 11 y(1) = -1 ) c1 + c2 = -1
x2 y’’ + x y’ – 9y = 0 (2)
y(1) = -1; y’(1) = 15 (3)
y = c1 x3 + c2 x-3
Example 1 (continued)
y(1) = -1 ) c1 + c2 = -1
y’(1) = 15 ) 3 c1 – 3 c2 = 15
Thus,
3 c1 + 3 c2 = -3
3 c1 – 3 c2 = 15
6 c = 12
) c1 = 2 and c2 = -3
Therefore, y = 2 x3 – 3 x-3 is the unique solution to (2), (3)!! 11

12. 4.3: Linear Independence
Definition: The functions {f1, f2, … , fn } are said to linearly dependent on a  x  b if there exist constants c1, c2, … , cn not all zero, such that
c1 f1(x) + c2 f2(x) + … + cn fn(x) = 0 (1)
for all x 2 [a, b].
If no such relation exists, we say {f1, f2, … , fn } are linearly independent on a  x  b. 12

13. Notes on Linear Independence
c1 f1(x) + c2 f2(x) + … + cn fn(x) = 0 (1)
Notes on Linear Independence
To say {f1, f2, … , fn} are linearly independent on [a,b] means that (1) holds for all x 2 [a,b] only if c1 = c2 = … = cn = 0.
To show {f1, f2, … , fn} are linearly independent on [a,b], a convenient approach is as follows:
Assume (1) holds for all x 2 [a,b] and show this forces c1 = c2 = … = cn = 0. 13

14. Notes on Linear Independence
c1 f1(x) + c2 f2(x) + … + cn fn(x) = 0 (1)
Notes on Linear Independence
{f1, f2, … , fn} are linearly dependent on [a,b] if and only if at least one of them is a linear combination of the others.
{f1, f2, … , fn} are linearly independent on [a,b] if and only if none of them is a linear combination of the others. 14

15. c1 f1(x) + c2 f2(x) + … + cn fn(x) = 0 (1)
Some Examples
Example 1: {sin2x, cos2x, 1} are linearly dependent on any real interval, as sin2x + cos2x – 1 = 0, for all x.
Example 2: {1, x, x2} are linearly independent on any real interval, since
c1 + c2 x + c3 x2 = 0 for all x 2 [a,b] ) c1 = c2 =c3 = 0 must hold.
Reason: Any polynomial of degree two has at most two distinct real roots (Fundamental Theorem of Algebra).
Hence, the only way to have c1 + c2 x + c3 x2 = 0 for all x 2 [a,b] is if the coefficients are all zero. 15