# L11 - Fall 2003 Math 308/501502 1 Introduction 1.1...

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Fall 2003 Math 308/501–502 1 Introduction 1.1 Background Mon, 01/Sep c 2003, Art Belmonte Summary When modeling applications, equating the rate at which a quantity changes (a derivative) with an application-specific way of formulating the rate of change leads to a differential equation (DE) ; i.e., an equation that contains some derivative(s) of an unknown function. Recall the concept of proportionality . With k a constant (often positive), we have the following mathematical relationships (among many). MATH ENGLISH y = kx y is proportional to x y = kx 2 y is proportional to the square of x y = kxz y is proportional to the product of x and z y = k / x 3 y is inversely proportional to the cube of x d P / dt = kP the rate of change of P is proportional to P Here are some terms used in the study of differential equations. independent variable : a variable in a DE upon which the unknown function depends; often t (time) or x (when the context is geometrical). dependent variable : a variable whose derivative appears in a DE; usually y , occasionally x or some other letter. coefficients : multipliers of the unknown function or its derivatives, either constant or depending on independent variable(s) only. ordinary differential equations (ODEs) : differential equations containing only ordinary derivatives.

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