Sec_1.1 - Ch I Introduction to Differential Equations 1.1...

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Ch. I Introduction to Differential Equations 1.1 Basic Definitions and Terminology Definition 1.1 Differential Equation An equation containing the derivatives or differentials of one or more dependent variables, with respect to one or more independent variables, is said to be a differential equation. Differential Equations are classified according to its type, order, and linearity. Type: I. Ordinary Differential Equation a. ( 29 4 0 y x dx xdy - + = b. du dv x dx dx - = c. 2 2 2 6 0 d y dy y dx dx - + = II. Partial Differential Equation a. 2 2 2 2 0 u u x y x + = - Laplace Equation or Harmonic Equation b. 2 2 2 2 2 u u k t x x = - Wave Equation c. u v y x x = - Order: The order of a differential equation is the highest-order derivative. a. 3 2 2 5 4 x d y dy y e dx dx + - = - 2 nd order. b. 4 2 2 4 2 0 u u k x t x + = - 4 th order. c. 2 2 , , , , ...... , 0 n n dy d y d y F x y dx dx dx = - general nth order.
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Linearity: 1. Linear: An equation is said to be linear if it can be written as ( 29 ( 29 ( 29 ( 29 ( 29 1 1 1 0 1 ........
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