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## Homework Statement

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

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