midterm_1_review (dragged)

midterm_1_review (dragged) - , such as time t ; and which...

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MA 36600 MIDTERM #1 REVIEW Chapter 1 § 1.1: Some Basic Mathematical Models; Direction Fields. Newton’s 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.” Newton’s 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 Mm r 2 = F = GMm r 2 for some (universal) constant G .I f M and r are the mass and radius of the Earth, respectively, then F = mg in terms of g = G M/r 2 =9 . 81 m / sec 2 = 32 . 5ft / sec 2 . In general say that we have an equation in the form dy dx = G ( x, y ) . A direction Feld or a slope Feld 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 diFerential equation to model a physical situation: #1. Identify the key variables. Decide which variables are
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Unformatted text preview: , 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 diFerential equation. #3. Identify the initial conditions. An initial value problem consists of (1) a diFerential equation and (2) a list of initial conditions. 1.2: Solutions of Some Dierential Equations. We review how to solve an initial value problem in the form dy dt = b + a y DiFerential 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 = a t + C 1 for some constant C 1 . Upon exponentiating both sides we nd that y ( t ) b/a = C 2 e at for some constant C 2 = e C 1 . or the initial condition, we set t = 0: C 2 = y b a = y ( t ) = b a + y b a e at . 1...
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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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