notes 3-2 - SECTION 3.2 A BIT OF THEORY FOR SECOND ORDER...

Info icon This preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
SECTION 3.2 A BIT OF THEORY FOR SECOND ORDER LINEAR ODE’S To answer the questions at the end of the last section we need a bit of theory, and we may as well do this for the general second order homogeneous linear ODE y 00 + p ( t ) y 0 + q ( t ) y = 0 . Notice that the coefficient of y 00 is 1. We abbreviate the left-hand-side as L [ y ], and then we call L a differential operator. If we let D stand for “take the derivative of” and then D 2 stand for “take the second derivative of,” and so on, we can even write L = D 2 + p ( t ) D + q ( t ), though we must use this notation carefully because it mixes up algebraic multiplication and application of the operator D . EXISTENCE AND UNIQUENESS FACT. This is carefully stated on page 144. The heart of the matter is that we need continuity in an interval and initial values for both y and y 0 at the same t 0 in the interval. Then the solution of the initial value problem exists and is unique and is defined throughout the interval on which we have continuity. We do not need
Image of page 1

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

View Full Document Right Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    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.

    Student Picture

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

  • Left Quote Icon

    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.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    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.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern