Elementary Differential Equations 8th edition by Boyce ch02

# Elementary Differential Equations, with ODE Architect CD

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—————————————————————————— —— CHAPTER 2. ________________________________________________________________________ page 18 Chapter Two Section 2.1 1 a b + Þ a b , Þ Based on the direction field, all solutions seem to converge to a specific increasing function. a b a b a b - Þ > 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 " a b 2 a b + Þ a b , . All slopes eventually become positive, hence all solutions will increase without bound. a b a b a b - Þ > œ / C > œ > / Î\$  - / Þ The integrating factor is , and hence It is . #> \$ #> #> evident that all solutions increase at an exponential rate. 3 a b +

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—————————————————————————— —— CHAPTER 2. ________________________________________________________________________ page 19 a b a b , C > œ " Þ . All solutions seem to converge to the function ! a b a b a b - Þ > œ / C > œ > / Î#  "  - / Þ The integrating factor is , and hence It is . #> # > > clear that all solutions converge to the specific solution . C > œ " ! a b 4 . a b + a b , . Based on the direction field, the solutions eventually become oscillatory. a b a b - Þ > œ > The integrating factor is , and hence the general solution is . C > œ =38 #>  \$-9= #> \$ - %> # > a b a b a b in which is an arbitrary constant. As becomes large, all solutions converge to the - > function C > œ \$=38 #> Î# Þ " a b a b 5 . a b +
—————————————————————————— —— CHAPTER 2. ________________________________________________________________________ page 20 a b , . All slopes eventually become positive, hence all solutions will increase without bound. a b a b a b ' - Þ > œ /B:  #.> œ / Þ The integrating factor is The differential equation . #> can be written as , that is, Integration of both / C  #/ C œ \$/ / C œ \$/ Þ #> w #> > #> > w a b sides of the equation results in the general solution It follows that C > œ  \$/  - / Þ a b > #> all solutions will increase exponentially. 6 a b + a b a b , Þ C > œ ! Þ All solutions seem to converge to the function ! a b a b - Þ > œ > The integrating factor is , and hence the general solution is . # C > œ  -9= > =38 #> - > > > a b a b a b # # in which is an arbitrary constant. As becomes large, all solutions converge to the - > function C > œ ! Þ ! a b 7 . a b +

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—————————————————————————— —— CHAPTER 2. ________________________________________________________________________ page 21 a b a b , Þ C > œ ! Þ All solutions seem to converge to the function ! a b a b a b a b - Þ > œ /B: > C > œ > /  - / Þ The integrating factor is , and hence It is . # # > > # # clear that all solutions converge to the function . C > œ ! ! a b 8 a b + a b a b , Þ C > œ ! Þ All solutions seem to converge to the function ! a b a b a b c d a b - Þ > œ C > œ >+8 >  G Î Þ Since , the general solution is . a b a b "  > "  > # # # # " It follows that all solutions converge to the function . C > œ ! ! a b 9 a b + Þ
—————————————————————————— —— CHAPTER 2.

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