{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

review1sol - Mathematics 38 Tufts University Department of...

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

View Full Document Right Arrow Icon
Tufts University Mathematics 38 Department of Mathematics Spring 2006 Solutions to Review for Exam 1 PLEASE READ: This is meant to be a representative list of problems to help you prepare for the first midterm examination. It is not meant to be all-inclusive. There may be material on this review that is not on the midterm, and vice versa. 1. In your own words, define the following terms. In each case, illustrate the correct use of these terms for a differential equation of your choice. (a) dependent variable Solution to differential equations are functions that tell how one quantity varies with another. The output of such a function is called the dependent variable . For example, one solution to the differential equation dy dx = y (1) is y = e x . In this case, y is the dependent variable . . . (b) independent variable . . . and x is the independent variable . The independent variable is what is input to the function that solves the differential equation. The independent variable is the variable with respect to which the derivatives in the differential equation are taken. (c) order of a differential equation The order of a differential equation is the order of the highest derivative (of the dependent variable with respect to the independent variable) that appears in the equation. For example, the differential equation d 4 y dx 4 + d 2 y dx 2 2 + x 2 y = 0 (2) is fourth order because the highest derivative in it is a fourth derivative. (d) linear differential equation A differential equation is linear if each of its terms either (i) is proportional to the dependent variable or one of its derivatives, or (ii) does not contain the dependent variable or any of its derivatives. For example, the general n ’th-order linear differential equation can be cast in the form A n ( x ) d n y dx n + A n - 1 ( x ) d n - 1 y dx n - 1 + · · · + A 2 ( x ) d 2 y dx 2 + A 1 ( x ) dy dx + A 0 ( x ) y = B ( x ) . (3) Here we have put all terms that do not contain the dependent variable on the right. Note that the functions A j ( x ) (for j = 0 , . . . , n ) and B ( x ) can be arbitrarily complicated functions of the independent variable. For example, Eq. (1) is first- order, with A 1 ( x ) = 1, A 0 ( x ) = - 1, and B ( x ) = 0.
Image of page 1

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

View Full Document Right Arrow Icon
(e) nonlinear differential equation If an ordinary differential equation is not linear, then it is nonlinear . For example, Eq. (2) is a fourth-order nonlinear differential equation. (f) homogeneous linear differential equation A linear differential equation is homogeneous if there are no terms that do not contain the dependent variable or any of its derivatives. That is, the right-hand side, B ( x ), of the linear n ’th-order differential equation written above must be zero. A homogeneous, linear differential equation is then of the form A n ( x ) d n y dx n + A n - 1 ( x ) d n - 1 y dx n - 1 + · · · + A 2 ( x ) d 2 y dx 2 + A 1 ( x ) dy dx + A 0 ( x ) y = 0 . (4) For example, Eq. (1) is a homogeneous first-order equation because it has B ( x ) = 0.
Image of page 2
Image of page 3
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