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mathew0135

  • 2 years ago

Solve the initial value problem: x(dy/dx)+y(x) = 9y(x)^(2), y(1) = -1

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  1. mathew0135
    • 2 years ago
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    \[x \frac{ dy }{dx } + y(x) = 9y(x)^2 , y(1) =-1\] Just the equation a little neater

  2. Dido525
    • 2 years ago
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    I would solve for dy/dx and intergrate.

  3. UnkleRhaukus
    • 2 years ago
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    the ex's int he brackets are just indicating the independent variable right?\[x \frac{ \text dy }{\text dx } + y = 9y^2 , \]

  4. Dido525
    • 2 years ago
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    and we know y(1) = -1.

  5. mathew0135
    • 2 years ago
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    I believe so, that's just how its been written. I'll try solving for dy/dx and integrating then,

  6. Dido525
    • 2 years ago
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    So I would sub those in.

  7. hartnn
    • 2 years ago
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    u realize that you can separate the variables easily here ?

  8. mathew0135
    • 2 years ago
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    \[\frac{ dy }{ dx } = \frac{ 9y(x)^2-y }{ x }\] Not too familiar with these problems, but basically i need to integrate that, no?

  9. hartnn
    • 2 years ago
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    u bring all terms of one variable on one side of = sign, like this : \(\large \frac{1}{9y^2-y}dy=\frac{1}{x}dx\) then integrate

  10. hartnn
    • 2 years ago
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    can u integrate both sides now ?

  11. mathew0135
    • 2 years ago
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    doing that now

  12. hartnn
    • 2 years ago
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    take your time :)

  13. mathew0135
    • 2 years ago
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    think i got it, left an x over the y side so i confused my self. \[1 = \ln(1-9y)-\ln(y)\]

  14. hartnn
    • 2 years ago
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    u integrated y-variable correctly , but what about \(\int (1/x)dx\) its not =1

  15. hartnn
    • 2 years ago
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    i meant u should get something like this : \(\ln x=ln(1-9y)-lny+c\) then use logarithmic properties to simplify

  16. mathew0135
    • 2 years ago
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    Okay, catching on think i figured out what i did wrong when i integrated last anyway.

  17. mathew0135
    • 2 years ago
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    Maybe y = \[\frac{ 1 }{ x+9 }\] May be a final solution, just simplifying the equation?

  18. hartnn
    • 2 years ago
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    ln cx= ln |(1-9y)/y| cxy= (1-9y) now use y(1) = -1 to find c.

  19. hartnn
    • 2 years ago
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    u got this simplification ? ---->ln cx= ln |(1-9y)/y|

  20. mathew0135
    • 2 years ago
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    yup, \[C(-1)(1)=1-9(-1)\] \[-C=10\] \[C=-10\] \[10xy=(1-9y)\]

  21. hartnn
    • 2 years ago
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    -10xy =1-9y or 10xy-9y+1=0

  22. mathew0135
    • 2 years ago
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    ahh, forgot the negative, think i have enough to try a few more of these questions any way. Thank you. :)

  23. hartnn
    • 2 years ago
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    welcome ^_^

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