anonymous
  • anonymous
Write and then solve for y = f(x) the differential equation for the statement: "The rate of change of y with respect to x is inversely proportional to y^4."
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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chestercat
  • chestercat
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Michele_Laino
  • Michele_Laino
hint: two quantities, say A and B are inversely proportional if the subsequent condition olds: A*B = k where k is a constant
Michele_Laino
  • Michele_Laino
holds*
anonymous
  • anonymous
dy/dx = ky^4 ?

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Michele_Laino
  • Michele_Laino
no, since that formula expresses the direct proportionality
Michele_Laino
  • Michele_Laino
hint: we can write this: A= dy/dx, and B=y^4
anonymous
  • anonymous
dy/dx = k/y^4
Michele_Laino
  • Michele_Laino
correct!
anonymous
  • anonymous
okay and now int(dy y^4) = int(dx k) (y^5)/5 = (x^2)/2 + C
anonymous
  • anonymous
\[y = \sqrt[5]{\frac{ 5x^2 }{ 2 } + C}\]
Michele_Laino
  • Michele_Laino
I got this: \[\Large \begin{gathered} \int {{y^4}dy} = \int {kdx} \hfill \\ \hfill \\ \frac{{{y^5}}}{5} = kx + C \hfill \\ \end{gathered} \]
anonymous
  • anonymous
doesn't the right side turn into \[\frac{ kx^2 }{ 2 } + C\]
anonymous
  • anonymous
no wait i see what i did wrong
anonymous
  • anonymous
okay so then \[y = \sqrt[5]{5kx + C}\]
Michele_Laino
  • Michele_Laino
correct!
Michele_Laino
  • Michele_Laino
sorry! I had to specify this: \[\Large C \in \mathbb{R}\] since you are searching for a real funtion
anonymous
  • anonymous
thank you so much :D
Michele_Laino
  • Michele_Laino
:)
anonymous
  • anonymous
quick question what does the \[\epsilon \] mean
Michele_Laino
  • Michele_Laino
in general, in mathematical analysis, the mathematicians use \epsilon in order to indicate a small and positive quantity, which, in some case goes to zero
anonymous
  • anonymous
okay , double thank you :D
Michele_Laino
  • Michele_Laino
:)

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