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Unformatted text preview: 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 ( 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 ( x ) =- 1, and B ( x ) = 0. (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 ( x ) y = 0 . (4) For example, Eq. (1) is a homogeneous first-order equation because it has B ( x ) = 0....
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