s_18416 - CHAPTER 2 Chapter Two Section 2.1 1a b a,b Based...

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—————————————————————————— —— CHAPTER 2. ________________________________________________________________________ page 18 Chapter Two Section 2.1 1 ab Based on the direction field, all solutions seem to converge to a specific increasing function. >œ/ C>œ > Î $" Î */  -/ Þ The integrating factor is , and hence . $> #> $> It follows that all solutions converge to the function C > œ >Î$  "Î* Þ " 2 , . All slopes eventually become positive, hence all solutions will increase without bound. >/Î $ -/Þ The integrating factor is , and hence It is . #> $ #> #> evident that all solutions increase at an exponential rate. 3 +
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—————————————————————————— —— CHAPTER 2. ________________________________________________________________________ page 19 ab ,C > œ " Þ . All solutions seem to converge to the function ! >œ/ C>œ >/ Î #" -/ Þ The integrating factor is , and hence It is . #> # > > clear that all solutions converge to the specific solution . C>œ" ! 4. + , . Based on the direction field, the solutions eventually become oscillatory. > œ> The integrating factor is , and hence the general solution is . C > œ =38 #>  $-9= #> $ - %> # > a b in which is an arbitrary constant. As becomes large, all solutions converge to the -> function C > œ $=38 #> Î# Þ " 5. +
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—————————————————————————— —— CHAPTER 2. ________________________________________________________________________ page 20 ab , . All slopes eventually become positive, hence all solutions will increase without bound. a b ' >œ/ B : # . >œ/ Þ The integrating factor is The differential equation . #> can be written as , that is, Integration of both /C # /C œ $ / /Cœ $ / Þ #> w #> > #> > w sides of the equation results in the general solution It follows that C> œ$/-/ Þ ># > all solutions will increase exponentially. 6 + C > œ!Þ All solutions seem to converge to the function ! > œ> The integrating factor is , and hence the general solution is . # C> œ -9= > =38 #> - >> > a b ## in which is an arbitrary constant. As becomes large, all solutions converge to the -> function C>œ! Þ ! 7. +
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—————————————————————————— —— CHAPTER 2. ________________________________________________________________________ page 21 ab C > œ!Þ All solutions seem to converge to the function ! a b > œ/B:> C> œ>/ -/ Þ The integrating factor is , and hence It is . # # > > ## clear that all solutions converge to the function . C>œ! ! 8 + All solutions seem to converge to the function ! c d > œ C> œ>+8 > GÎ Þ Since , the general solution is . "> # # # # " It follows that all solutions converge to the function . ! 9
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—————————————————————————— —— CHAPTER 2. ________________________________________________________________________ page 22 ab , . All slopes eventually become positive, hence all solutions will increase without bound. ˆ‰ ' > œ/B: . > œ/ The integrating factor is . The differential equation can . " # >Î# be written as , that is, Integration / C  / CÎ# œ $> / Î# œ $> / Î#Þ >Î# w >Î# >Î# >Î# /C Î # >Î# w of both sides of the equation results in the general solution All C> œ$ >'-/ Þ >Î# solutions approach the specific solution C>œ$ > ' Þ !
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s_18416 - CHAPTER 2 Chapter Two Section 2.1 1a b a,b Based...

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