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Unformatted text preview: 1.1: SOME BASIC MATHEMATICAL MODELS KIAM HEONG KWA A differential equation is an equation between specified derivatives of an unknown function, its values, and known quantities and functions. The equation is called an ordinary differential equation (ODE) if the unknown is a func- tion of a single variable, and a partial differential equation (PDE) if the unknown depends on more than one independent variables. By an n th order ODE , one means an equation of the form (0.1) y ( n ) = F bracketleftbig t, y, y , , y ( n 1) bracketrightbig , where F is a function from a subset of R n +1 into R . Here y is the unknown of the equation and t is the independent variable. A solution of (0.1) is any function ( t ) defined on an open interval J R such that (0.2) ( n ) ( t ) = F bracketleftbig t, ( t ) , ( t ) , , ( n 1) ( t ) bracketrightbig for all t J . Tacit in the definition is the n times differentiability of the solution ( t ) on I . The graph of such a solution ( t ) is called an integral curve of (0.1). If, in addition to (0.1), the unknown y is required to satisfy a set of initial conditions (0.3) y ( t ) = y , y ( t ) = y , , y ( n 1) ( t ) = y ( n 1) , y ( n ) ( t ) = y ( n ) , where t , y , y , , y ( n 1) , and y ( n ) are prescribed quantities, then the problem of finding a solution ( t ) of (0.1) such that (0.4) ( t ) = y , ( t ) = y , , ( n 1) ( t ) = y ( n 1) , ( n ) ( t ) = y ( n ) is called an initial value problem (IVP) . Date : January 1, 2011. 1 2 KIAM HEONG KWA 1. First-Order Equations To further illustrate the definitions from the last section, consider the first-order equation (1.1) dy dt = f ( t, y ) , where f is an arbitrary function of two variables, together with an initial condition (1.2) y ( t ) = y . The function f is called the rate function and is usually assumed to be continuously differentiable in its domain J U , where J and U are some open intervals containing t and y respectively, in the sense that its partial derivatives exist and are continuous in J U . A function ( t ) is called a solution of (1.1) if (1.3) ( t ) = f ( t, ( t )) for all t I . Such a solution ( t ) is said to be a solution of the IVP...
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