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113_pdfsam_math 54 differential equation solutions odd

# 113_pdfsam_math 54 differential equation solutions odd -...

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Exercises 3.2 <<< >>>>> <<<<< 0 a/b Figure 3–A : The phase line for p 0 =( a bp ) p . Putting this value for k into the equation for p ( t )g ives p ( t ) = 1000 e ( t ln 3) / 7 = 1000 · 3 t/ 7 . To estimate the population in 2010 we plug t = 2010 1980 = 30 into this formula to get p (30) = 1000 · 3 30 / 7 110 , 868 splakes . 11. In this problem, the dependent variable is p , the independent variable is t , and the function f ( t, p )=( a bp ) p .S ince f ( t, p )= f ( p ), i.e., does not depend on t , the equation is autonomous. To Fnd equilibrium solutions, we solve f ( p )=0 ( a bp ) p =0 p 1 ,p 2 = a b . Thus, p 1 ( t ) 0and p 2 ( t ) a/b are equilibrium solutions. ±or p 1 <p<p 2 , f ( p ) > 0, and f ( p ) < 0when p>p 2 .(A l so , f ( p ) < 0for p<p 1 .) Thus the phase line for the given equation is as it is shown in ±igure 3-A. ±rom this picture, we conclude that the equilibrium p = p 1 is
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