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Unformatted text preview: MA 36600 MIDTERM #1 REVIEW Chapter 1 1.1: Some Basic Mathematical Models; Direction Fields. Newtons Second Law of Motion is the statement F = ma ; it really means m d 2 x dt 2 = the sum of the forces on the object. Newtons Law of Gravitational Attraction is the statement any body with mass M attracts any other body with mass m directly toward the mass M , with a magnitude proportional to the (product of the two) masses and inversely pro- portional to the square of the distance separating them. Another way to say this is F M m r 2 = F =- GM m r 2 for some (universal) constant G . If M and r are the mass and radius of the Earth, respectively, then F =- mg in terms of g = GM/r 2 = 9 . 81m / sec 2 = 32 . 5ft / sec 2 . In general say that we have an equation in the form dy dx = G ( x,y ) . A direction field or a slope field is a plot in the xy-plane where for each point ( x,y ) we plot an arrow with slope G ( x,y ). There are three key steps to using a differential equation to model a physical situation: #1. Identify the key variables. Decide which variables are independent , such as time t ; and which are dependent , such as y = y ( t ). #2. Articulate the principle that underlies the problem under investigation. This can be used to articulate the differential equation. #3. Identify the initial conditions. An initial value problem consists of (1) a differential equation and (2) a list of initial conditions. 1.2: Solutions of Some Differential Equations. We review how to solve an initial value problem in the form dy dt =- b + ay Differential Equation y (0) = y Initial Condition for some constants a and b : dy dt = a y- b a 1 y- b a dy dt = a d dt ln y- b a = a = ln y- b a = at + C 1 for some constant C 1 . Upon exponentiating both sides we find that y ( t )- b/a = C 2 e at for some constant C 2 = e C 1 . For the initial condition, we set t = 0: C 2 = y- b a = y ( t ) = b a + y- b a e at . 1 2 MA 36600 MIDTERM #1 REVIEW 1.3: Classification of Differential Equations. There are two types of differential equations: ODEs or ordinary differential equations are equations which do not involve partial derivatives. PDEs or partial differential equations are equations which do involve partial derivatives. Consider a function y = y ( t ), and let y ( k ) denote the k th derivative. An n th order differential equation is an ordinary differential equation in the form F t, y, y (1) , y (2) , ..., y ( n ) = 0 = y ( n ) = G t, y, y (1) , y (2) , ...,y ( n- 1) . We say that this differential equation is linear if G ( t,~ y ) = g ( t ) + g 1 ( t ) y 1 + g 2 ( t ) y 2 + + g n ( t ) y n is a linear function for the vector ~ y = ( y 1 ,y 2 ,...,y n ). Otherwise, we say that this differential equation is nonlinear ....
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- Spring '09
- Basic Math