section%201.1 - Chapter 1 Introduction to Differential...

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Unformatted text preview: Chapter 1. Introduction to Differential Equations 1. Differential Equations. An equation containing the derivative of one or more dependent variables, with respect to one or more independent variables, is said to be differential equation (DE). If the derivative is taken with respect to only one independent variable, the equation is said to be ordinary differential equation (ODE). If the equation contains partial derivatives, the equation is said to be partial differential equation (PDE). The order of a DE is the order of the highest derivative in the equation. 2. Solution of an ODE: Any function, φ, defined on an interval I and possessing at least n derivatives that are continuous on I , which when substituted into an nth-order ordinary differential equation reduces the equation to an identity, is said to be a solution of the equation on the interval. Systems of Differential Equations. A system of differential equations is two or more equations involving the derivatives of two or more unknown functions of a single independent variable. For example dx dt dy dt = f (t, x, y ) = g (t, x, y ) 4. Initial Value problem. dn y dxn = f (x, y, y , · · · , y (n−1) ) y (x0 ) = y0 , y (x0 ) = y1 , · · · , y (n−1) (x0 ) = yn−1 1 ...
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