ch07_02 - 7-13 SECTION 7.2 . Separable Differential...

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7-13 SECTION 7.2 . . Separable Differential Equations 577 argon-40 does not remain in molten lava. For a ratio of 0.00012, find the age of the rock. Comment on why this method is used to date very old rocks. 3. Three confused hunting dogs start at the vertices of an equi- lateral triangle of side 1. Each dog runs with a constant speed aimed directly at the dog that is positioned clockwise from it. The chase stops when the dogs meet in the middle (having grabbed each other by their tails). How far does each dog run? [Hints: Represent the position of each dog in polar coordinates ( r ) with the center of the triangle at the origin. By symme- try, each dog has the same r -value, and if one dog has angle θ , then it is aimed at the dog with angle θ 2 π 3 . Set up a differential equation for the motion of one dog and show that there is a solution if r ± ( θ ) = 3 r . Use the arc length formula L = ± θ 2 θ 1 ² [ r ± ( θ )] 2 + [ r ( θ )] 2 d θ. ] 4. To generalize exercise 3, suppose that there are n dogs starting at the vertices of a regular n -gon of side s .If α is the interior angle from the center of the n -gon to adjacent vertices, show that the distance run by each dog equals s 1 cos α . What hap- pens to the distance as n increases without bound? Explain this in terms of the paths of the dogs. 7.2 SEPARABLE DIFFERENTIAL EQUATIONS In section 7.1, we solved two different differential equations: y ± ( t ) = ky ( t ) and y ± ( t ) = k [ y ( t ) T a ] . These are both examples of separable differential equations. We will examine this type of equation at some length in this section. First, we consider the more general first-order ordinary differential equation y ± = f ( x , y ) . (2.1) Here, the derivative y ± of some unknown function y ( x )isgiven as a function f of both x and y . Our objective is to find some function y ( x )(a solution ) that satisfies equation (2.1). The equation is first-order, since it involves only the first derivative of the unknown function. We will consider the case where the x ’s and y ’s can be separated. We call equation (2.1) separable if we can separate the variables, i.e., if we can rewrite it in the form g ( y ) y ± = h ( x ) , where all of the x ’s are on one side of the equation and all of the y ’s are on the other side. EXAMPLE 2.1 A Separable Differential Equation Determine whether the differential equation y ± = xy 2 2 is separable. Solution Notice that this equation is separable, since we can rewrite it as y ± = x ( y 2 2 y ) and then divide by ( y 2 2 y ) (assuming this is not zero), to obtain 1 y 2 2 y y ± = x . ± NOTE Do not be distracted by the letter used for the independent variable. We frequently use the independent variable x ,asin equation (2.1). Whenever the independent variable represents time, we use t as the independent variable, in order to reinforce this connection, as we did in example 1.2. There, the equation describing radioactive decay was given as y ± ( t ) = ( t ) .
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578 CHAPTER 7 . .
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ch07_02 - 7-13 SECTION 7.2 . Separable Differential...

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