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experimentX

  • 2 years ago

solve: \[ x\; {\partial z \over \partial x} + y\; {\partial z \over \partial y} = z \] that passes though \( x^2+y^2+z^2=25 \) and \( x+y=1 \)

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  1. experimentX
    • 2 years ago
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    the answer according to book is \[ 25(x+y) = x^2+y^2+z^2\] looks like intersection of two surfaces more than solution of DE

  2. experimentX
    • 2 years ago
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    the solution looks like this ...

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  3. experimentX
    • 2 years ago
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    what kind of surface is that ... man i thought it would be a circle.

  4. mukushla
    • 2 years ago
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    man how can we solve something like this? from the \(x^2+y^2+z^2=25\) we have \[{\partial z \over \partial x}=-\frac{x}{z}\]\[{\partial z \over \partial y}=-\frac{y}{z}\]put back in the original equation\[x^2+y^2=-z^2\]lol

  5. experimentX
    • 2 years ago
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    i'm supposed to do it by charpit's method ... an charpit's method is supposed to be easy than lagrange's method ... lol

  6. hartnn
    • 2 years ago
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    i exactly did the same thing and got the same result!! x^2+y^2+z^2=0 !!

  7. mukushla
    • 2 years ago
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    but this is wrong because it gives x=y=z=0

  8. experimentX
    • 2 years ago
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    The lagrange method works ... but this is too ugly ... I have to eliminate z from 4th order equation

  9. experimentX
    • 2 years ago
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    this is my last problem from first order DE ... I'm moving on to second order DE after this Q http://www.mathresources.com/products/mathresource/maa/charpits_method.html

  10. mukushla
    • 2 years ago
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    man i cant get that answer with charpit

  11. mukushla
    • 2 years ago
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    @experimentX santosh where are u....lol

  12. experimentX
    • 2 years ago
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    still here man ... fishing answer from another site.

  13. experimentX
    • 2 years ago
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    what did you get for answer? i got z = a x + phi(a) y + c

  14. experimentX
    • 2 years ago
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    had been doing \[ z = k_1x+k_2y+k_3 \]

  15. experimentX
    • 2 years ago
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    k2 should be some function of k1 ... i guess there it would reduce my trouble by half

  16. mukushla
    • 2 years ago
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    man let me try again..

  17. experimentX
    • 2 years ago
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    let me try it again too

  18. Valpey
    • 2 years ago
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    The intersection of the sphere and the plane will be a curve. The equation of the curve will solve \[x^2+(1-x)^2+z^2=25\]

  19. experimentX
    • 2 years ago
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    I need to find the particular integral of the given DE passing through both of these curves.

  20. experimentX
    • 2 years ago
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    no luck ... can't find \( \phi(k_1) \)

  21. Valpey
    • 2 years ago
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    \[ \frac{\partial z}{\partial{x}}=\frac{2x-2(1-x)}{-2z}=\frac{2x-1}{-z}\]Similarly solve for y and \[ \frac{\partial y}{\partial{z}}\]

  22. experimentX
    • 2 years ago
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    and substitute it there?

  23. Valpey
    • 2 years ago
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    Sure

  24. experimentX
    • 2 years ago
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    Looks like I messed up with Lagrange sol ... it didn't seem that difficult \[z/x = \phi(y/x)\]

  25. experimentX
    • 2 years ago
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    the problem reduced to \[ z = ax +by \\ x^2+y^2+z^2=25\\ x+y=1\] Need to eliminate 'a' and 'b' from these equations.

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