{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Homog_DEs_Results

# Homog_DEs_Results - 14 Homogeneous Linear Equations The Big...

This preview shows pages 1–3. Sign up to view the full content.

14 Homogeneous Linear Equations — The Big Theorems Let us continue the discussion we were having at the end of section 12.3 regarding the general solution to any given homogeneous linear differential equation. By then we had seen that any linear combination of particular solutions, y ( x ) = c 1 y 1 ( x ) + c 2 y 2 ( x ) + · · · + c M y M ( x ) , is another solution to that homogeneous differential equation. In fact, we were even beginning to suspect that this expression could be used as a general solution to the differential equation provided the y k ’s were suitably chosen. In particular, we suspected that the general solution to any second-order, homogeneous linear differential equation can be written y ( x ) = c 1 y 1 ( x ) + c 2 y 2 ( x ) were c 1 and c 2 are arbitrary constants, and y 1 and y 2 are any two solutions that are not constant multiples of each other. These suspicions should have been reinforced in the last chapter in which general solutions were obtained via reduction of order. In the examples and exercises, you should have noticed that the solutions obtained to the given homogeneous differential equations could all be written as just described. It is time to confirm these suspicions, and to formally state the corresponding results. These results will not be of merely academic interest. We will use them for much of the rest of this text. For practical reasons, we will split our discussion between this and the next chapter. This chapter will contain the statements of the most important theorems regarding the solutions to homogeneous linear differential equations, along with a little discussion to convince you that these theorems have a reasonable chance of being true. More convincing (and lengthier) analysis will be carried out in the next chapter. 14.1 Preliminaries and a Little Review We are discussing general homogeneous linear differential equations. If the equation is of second order, it will be written as ay ′′ + by + cy = 0 . 297

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
298 Homogeneous Linear Equations — The Big Theorems More generally, it will be written as a 0 y ( N ) + a 1 y ( N 1 ) + · · · + a N 2 y ′′ + a N 1 y + a N y = 0 where N , the order, is some positive integer. The coefficients — a , b and c in the second order case, and the a k ’s in the more general case — will be assumed to be continuous functions over some open interval I , and the first coefficient — a or a 0 — will be assumed to be nonzero at every point in that interval. Recall the “principle of superposition”: If { y 1 , y 2 , . . . , y K } is a set of particular solutions over I to a given homogeneous linear equation, then any linear combination of these solutions, y ( x ) = c 1 y 1 ( x ) + c 2 y 2 ( x ) + · · · + c K y K ( x ) for all x in I , is also a solution over I to the the given differential equation. Also recall that this set of y ’s is called a fundamental set of solutions (over I ) for the given homogeneous differential equation if and only if both of the following hold: 1. The set is linearly independent over I (i.e., none of the y k ’s is a linear combination of the others over I ).
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern