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anonymous

  • one year ago

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."

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  1. Michele_Laino
    • one year ago
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    hint: two quantities, say A and B are inversely proportional if the subsequent condition olds: A*B = k where k is a constant

  2. Michele_Laino
    • one year ago
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    holds*

  3. anonymous
    • one year ago
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    dy/dx = ky^4 ?

  4. Michele_Laino
    • one year ago
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    no, since that formula expresses the direct proportionality

  5. Michele_Laino
    • one year ago
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    hint: we can write this: A= dy/dx, and B=y^4

  6. anonymous
    • one year ago
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    dy/dx = k/y^4

  7. Michele_Laino
    • one year ago
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    correct!

  8. anonymous
    • one year ago
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    okay and now int(dy y^4) = int(dx k) (y^5)/5 = (x^2)/2 + C

  9. anonymous
    • one year ago
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    \[y = \sqrt[5]{\frac{ 5x^2 }{ 2 } + C}\]

  10. Michele_Laino
    • one year ago
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    I got this: \[\Large \begin{gathered} \int {{y^4}dy} = \int {kdx} \hfill \\ \hfill \\ \frac{{{y^5}}}{5} = kx + C \hfill \\ \end{gathered} \]

  11. anonymous
    • one year ago
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    doesn't the right side turn into \[\frac{ kx^2 }{ 2 } + C\]

  12. anonymous
    • one year ago
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    no wait i see what i did wrong

  13. anonymous
    • one year ago
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    okay so then \[y = \sqrt[5]{5kx + C}\]

  14. Michele_Laino
    • one year ago
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    correct!

  15. Michele_Laino
    • one year ago
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    sorry! I had to specify this: \[\Large C \in \mathbb{R}\] since you are searching for a real funtion

  16. anonymous
    • one year ago
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    thank you so much :D

  17. Michele_Laino
    • one year ago
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    :)

  18. anonymous
    • one year ago
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    quick question what does the \[\epsilon \] mean

  19. Michele_Laino
    • one year ago
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    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

  20. anonymous
    • one year ago
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    okay , double thank you :D

  21. Michele_Laino
    • one year ago
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    :)

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