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gerryliyana

  • 2 years ago

Solve dy/dx = (x + 4y -1)^2

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  1. gerryliyana
    • 2 years ago
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    solve \(\frac{ dy }{ dx } = (x + 4y -1)^{2}\)

  2. carson889
    • 2 years ago
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    It is 2x + 8y - 2.

  3. carson889
    • 2 years ago
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    Got it by expanding the brackets: x^2 + 4yx - x + 4xy + 16y^2 -4y -x - 4y + 1. Which simplifies to: x^2 + 8xy + 16y^2 - 2x -8y + 1. Taking the derivative of that with respect to x: 2x + 8y + 0 - 2 - 0 + 0 Which simplifies to 2x + 8y -2 or 2(x+4y-1)

  4. carson889
    • 2 years ago
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    Or using chain rule: \[2(x+4y-1)^{1}(\frac{ d }{ dx }(x+4y-1))\] \[\frac{ d }{ dx } = 1\] Therfore, \[2(x+4y-1)(1) = 2x + 8y - 2\]

  5. Idealist
    • 2 years ago
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    2(x+4y-1)=2x+8y-2.

  6. gerryliyana
    • 2 years ago
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    ok thank all a lots i appreciate it :)

  7. Idealist
    • 2 years ago
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    No Problem.

  8. gerryliyana
    • 2 years ago
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    @carson889: wait i mean \[\frac{ dy }{ dx } = (x+4y - 1)^{}\] \((x+4y -1)^{2}\) is the result of \(\frac{ dy }{ dx }\) it isn't \[\frac{ d }{ dx} (x +4y -1)^{2}\] any idea???

  9. LolWolf
    • 2 years ago
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    You need to use the chain rule, as this is implicit differentiation, not partial.

  10. gerryliyana
    • 2 years ago
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    what did u get it??

  11. LolWolf
    • 2 years ago
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    So, are we trying to find \(y\) ?

  12. LolWolf
    • 2 years ago
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    (As a function of \(x\) )

  13. gerryliyana
    • 2 years ago
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    it seems like http://openstudy.com/study#/updates/50a9b0e8e4b064039cbd16a2

  14. LolWolf
    • 2 years ago
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    Yeah, so it's an actually differential equation which you have to solve. Here:

  15. LolWolf
    • 2 years ago
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    Define (all in terms of x): \[ v=4y+x\implies v'=4y'+1 \]So: \[ \frac{v'-1}{4}=(v-1)^2\implies\\ \frac{v'}{4v^2-8v+5}=1 \]Integrating both sides: \[ \int \frac{v'}{4v^2-8v+5}\;dx=\int 1\;dx=\\ \frac{\tan^{-1}(2v-2)}{2}=x+C \]Solving for \(v\): \[ v=\frac{\tan(2(x+C))+2}{2} \]Substituting \(y\): \[ y=\frac{\tan(2(x+C))-2x+2}{8} \] That took forever.

  16. gerryliyana
    • 2 years ago
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    awesome..., thank u @LolWolf

  17. LolWolf
    • 2 years ago
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    Sure thing

  18. gerryliyana
    • 2 years ago
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    @LolWolf

  19. gerryliyana
    • 2 years ago
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    i tried to calculate on Maple 13, but, look at the picture below

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  20. gerryliyana
    • 2 years ago
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    is not equal to (x+4y-1)^2

  21. LolWolf
    • 2 years ago
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    This is the equation I get, when solving the differential with Mathematica:

  22. LolWolf
    • 2 years ago
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    In real form:

  23. gerryliyana
    • 2 years ago
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    ok...,

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  24. gerryliyana
    • 2 years ago
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    it is right, isn't it?

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