Deriving maxwell equations to obtain plane wave The Maxwell Equation is written the below form .
**IN VACUUM **
$$\mathbf{\nabla \cdot E} = 0$$ 
$$ \mathbf{\nabla \times E} = - \frac{\partial \mathbf{B}}{\partial t}$$
$$\mathbf{\nabla \cdot B} = 0$$
**IN VACUUM **
$$\mathbf{\nabla \times B} = \frac{1}{c^2}\frac{\partial \mathbf{E}}{\partial t}$$
By organizing the structure we can get ,
$$\nabla^2\overrightarrow{\mathbf{E}} =  \frac{1}{c^2}\frac{\partial ^2 \mathbf{E}}{\partial t^2}$$
In order to obtain the relationship between the electric field and magnetic field also to prove the plane wave condition , we set the  electric field of the $y$ and $z$ dimension to zero. 
$\require{cancel}$
$$\overrightarrow{\mathbf{E}}\rightarrow\langle E_x,0,0\rangle$$
Thus, we obtain the fact that the Laplacian operator multiplied by the electric field will result in the second partial derivative of $E_x$ with respect to $z$, or $$\nabla^2\overrightarrow{\mathbf{E}} =  \frac{\partial^2}{\partial z^2}E_x$$ However, why shouldn't it have two terms on the right side, that is the second partial derivative of $E_x$ with respect to $z$ plus the second partial derivative of  $E_x$ with respect to $y$:
$$\nabla^2\overrightarrow{\mathbf{E}}=\frac{\partial^2}{\partial z^2}E_x+\frac{\partial^2}{\partial y^2}E_x$$
$$\mu_0\epsilon_0\frac{\partial^2}{\partial t^2}E_x = \left(\frac{\partial^2}{\partial z^2}E_x+\cancel {\frac{\partial^2}{\partial y^2}E_x}\right)$$
Another problem is that when you take the curl of this electric field you would also result in only one term of partial derivative of $E_x$ with respect to $z$ instead of two terms. I.e., it's this:
$$\nabla\times\overrightarrow{\mathbf{E}} = \frac{\partial E_x}{\partial z}\hat{y}+\cancel{\left(-\frac{\partial E_x}{\partial y}\right)\hat{z}}$$
But why aren't the crossed out terms present?
 A: You can arbitrarily rotate your coordinate system so that the electromagnetic wave of traveling in the z direction. With that restriction, you still have the option to rotate your coordinate system around the z axis do that the electromagnetic field has no y component.
A: Presumably because you're using a source that's been purposefully oversimplified as a pedagogical device. It's impossible to tell exactly what you're doing from the question as currently posed, and you've skipped the crucial step of referencing your sources (which presumably you've been told is essential at multiple points), which cripples your question's answerability, but the core remains: at no point is it essential to assume that the electric field points only in one dimension, or has dependence only on one cartesian variable, to prove the core equations of electromagnetism.
In particular, it appears that you're not trying to derive any of the Maxwell equations, but instead you're trying to derive the wave equation from the Maxwell equations. The full derivation is proved in any suitably advanced EM textbook (say, Griffiths and above) and it does not require any specific relationship between the electromagnetic fields and the coordinate system.
