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ELEMENTRY DIFFERENTIAL EQUATIONS 2.5

# ELEMENTRY DIFFERENTIAL EQUATIONS 2.5 - ELEMENTRY...

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ELEMENTRY DIFFERENTIAL EQUATIONS 9 th Edition, Boyce Chapter 2.5 5- 7- (a) Consider the equation d y /d t = k (1 - y ) 2 , (Equation A) where k is a positive constant. Show that y = 1 is the only critical point, with the correspopnding equilibrium solution φ( t ) = 1.

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Solution: Denote the function as f( y ) = k (1 - y ) 2 . The critical points y p are a reference to f(y p ) = 0, so the only critical point is y p = 1. Critical points are also a reference to equilibrium solutions φ( t ) = y p . (φ( t ) = 1 is such a solution.) (b) Sketch f( y ) versus y . Show that y is increasing as a function of t for y < 1 and also for y > 1. The phase line has upward-shifting arrows both below and above y = 1. Thus solutions below the equilibrium solution approach it, and those above it grow farther away. Therefore φ( t ) = 1 is semistable. Solution: Graph of f( y ) = k (1 - y ) 2 : Let y = φ( t ) be a solution to the differential equation y = f( y ). When y ≠ 1, f( y ) > 0 (the derivative φ
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ELEMENTRY DIFFERENTIAL EQUATIONS 2.5 - ELEMENTRY...

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