Lecture_3(dragged) - MA 36600 LECTURE NOTES FRIDAY JANUARY 16 3 If we did not keep track of the initial condition y(0 = y0 we would call the

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Unformatted text preview: MA 36600 LECTURE NOTES: FRIDAY, JANUARY 16 3 If we did not keep track of the initial condition y (0) = y0 , we would call the solution of the differential equation alone the general solution. For example, y (t) = (b/a) + C1 eat would be the general solution for the differential equation above because it involves an arbitrary constant C1 . If we do keep track of the initial condition, we would call the solution the particular solution. For example, y (t) = (b/a) + (y0 − b/a) eat would be the particular solution corresponding to y (0) = y0 . Classification of Differential Equations Types of Differential Equations. There are two: • ODEs or ordinary differential equations are differential equations which do not involve partial derivatives. • PDEs or partial differential equations are differential equations which do involve partial derivatives. Each of the examples of differential equation we have seen are types of ODEs, so here is an example of a PDE: Say that we have a rod of length L sitting in or above a heat source. (For example, the rod might have one end in a furnace.) We wish to compute the temperature u = u(x, t) as a position x (where 0 ≤ x ≤ L) and a time t. Note that both time t and position x are independent variables, and u = u(x, t) is a dependent variable. The differential equation which models this physical situation is called the Heat Equation : ∂2u ∂u = ∂ x2 ∂t for some nonzero constant α which depends on the characteristics of the rod. In this course, we will only be interested in ordinary differential equations. α2 Order of an ODE. Consider a function y = y (t). Let y (k) denote the k th derivative. For example, dy d2 y , y (2) = 2 , dt dt An ordinary differential equation is an expression in the form ￿ ￿ F t, y, y (1) , y (2) , . . . , y (n) = 0 y (0) = y, y (1) = ··· for some function F . The largest order n of derivative y (n) which appears in this expression is called the order of the differential equation. We give a few of examples. Let x = x(t) denote the position of a mass m at time t under the influence of gravity and air resistance. If this mass is near sea level (at the surface of the Earth), Newton’s Law of Gravity states dx d2 x + m 2 = 0. dt dt This is an ordinary differential equation of second order because it involves the second derivative of the position x = x(t). If instead we use the velocity v = x￿ (t), we find the differential equation dv mg + γ v + m = 0. dt This is an ordinary differential equation of first order because it involves just the first derivative of the velocity v = v (t). Let P = P (t) denote the size of a population at time t. If we assume that the rate of change of the size if proportional both to the size of the population as well as the difference from the maximal sustainable size K of the population, then we find the differential equation ￿ ￿ P dP −r P 1 − + = 0. K dt This is an ordinary differential equaiton of first order because it involves just the first derivative of the population size P = P (t). mg + γ ...
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This note was uploaded on 11/30/2011 for the course MATH 366 taught by Professor Edraygoins during the Spring '09 term at Purdue University-West Lafayette.

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