**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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