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Help solving (complex) simultaneous equations

  1. Apr 17, 2009 #1

    KFC

    User Avatar

    1. The problem statement, all variables and given/known data
    I am trying to solving the following complex equation for [tex]x[/tex] and [tex]\theta[/tex]

    [tex]a\sinh(2x) e^{-i\theta} + y\sinh^2x e^{-i2\theta} + y^*\cosh^2(x) = 0[/tex]

    where [tex]a[/tex] is real constant, [tex]x[/tex] and [tex]\theta[/tex] are also real parameter. [tex]y[/tex] is complex number, [tex]y^*[/tex] is the complex conjugate.

    Solve for [tex]x[/tex] and [tex]\theta[/tex] (in terms of y and a)

    2. The attempt at a solution
    Let
    [tex]y = |y| e^{i\varphi}[/tex]

    and multiply the equation with [tex]y[/tex]

    [tex]ay\sinh(2x) e^{-i\theta} + y^2\sinh^2x e^{-i2\theta} + |y|^2\cosh^2(x) = 0[/tex]

    Now let the real part and imaginary part equals ZERO.

    [tex]
    \begin{cases}
    a\sinh(2x) |y|\cos(\theta-\varphi) + |y|^2\sinh^2(x)\cos(2\theta-2\varphi) + |y|^2\alpha^2 = 0, \\[3.8mm]
    a\sinh(2x) |y|\sin(\theta-\varphi) + |y|^2\sinh^2(x)\sin(2\theta-2\varphi) = 0
    \end{cases}
    [/tex]

    I tryied to solve that two days ago, I tried many way to simpliy that but still find no way to get the soluton. Could anyone give me some hints?

    Thanks
     
    Last edited: Apr 18, 2009
  2. jcsd
  3. Apr 18, 2009 #2

    Mark44

    Staff: Mentor

    This doesn't make sense to me. How can you solve one equation for two variables? This seems to me like asking someone to solve y = 2x for x and y. You can solve the equation for y in terms of x, or you can solve for x in terms of y, but you can't solve it simultaneously for both variables.
     
  4. Apr 18, 2009 #3

    KFC

    User Avatar

    How come. This is a equation for complex variable, the real part and imaginary part gives two equations.
     
  5. Apr 18, 2009 #4

    Mark44

    Staff: Mentor

    OK, I see.
    [tex] a\sinh(2x) |y|\sin(\theta-\varphi) + |y|^2\sinh^2(x)\sin(2\theta-2\varphi) = 0[/tex]
    In your 2nd equation, I think the sine arguments should be the other way around. Also, you can expand [itex]sin(2(\phi - \theta))[/itex] as [itex]2sin(\phi - \theta) cos(\phi - \theta)[/itex]. Then you'll have one factor the same in both terms, which hopefully leaves you with the other factor that you can do something with.
     
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