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Unformatted text preview: 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 ∝ 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 xyplane 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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This note was uploaded on 05/24/2011 for the course MA 36600 taught by Professor Ma during the Spring '09 term at Purdue.
 Spring '09
 MA
 Basic Math

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