I am trying to find a solution to a system of the following two differential equations:
$$\frac{\partial^2 B}{\partial x^2} = -(\mu_0 \epsilon_0)^2 \frac{\partial^2 B}{\partial t^2}\tag1$$
$$\frac{\partial^2 E}{\partial x^2} = \mu_0 \epsilon_0 \frac{\partial^2 B}{\partial t^2}\tag2$$
(To some of those who know physics, these equations may seem erroneous, but that is intentional.)
Now, by noting that the general solution to the wave equation given by $(1)$ is of the form
$$B(x,t) = F\left(x-\frac{it}{\mu_0 \epsilon_0}\right)+G\left(x+\frac{it}{\mu_0 \epsilon_0}\right)$$
and by using Euler's identity and the principle of superposition, I obtained the following solution for $B(x,t)$:
$$B = e^{-x}f(t)+e^{-\frac{t}{\mu_0 \epsilon_0}} g(x)$$
where
$$f(t) = C_1 \cos \left(\frac{t}{\mu_0 \epsilon_0}\right) + C_2 \sin \left(\frac{t}{\mu_0 \epsilon_0}\right)\\ g(x) = C_3 \cos x + C_4 \sin x$$
and $C_i$ are arbitrary constants. It follows by equation $(2)$ that
$$E = -\frac{1}{\mu_0 \epsilon_0} B$$
I was told that these solutions are almost correct, but apparently plugging my solutions back into "equations for E and B" (I am not certain whether these really refer to the differential equations) will give me some information about $C_1,C_2,C_3,C_4$. However, obviously, plugging the solutions into the differential equations simply gives me identities and no new information. The only ways I can think of to determine relationships of the coefficients are initial or boundary conditions, but there is none for this problem.
Are there any other ways to determine relationships for the coefficients? Thank you very much.
P.S. In case this helps, below are equations available for me to use to solve this problem, if necessary (but they may be useless in one-dimensional space):
$$\mathbf{\nabla} \cdot \mathbf{E} = 0\\ \mathbf{\nabla} \cdot \mathbf{B} = 0\\ \mathbf{\nabla} \times \mathbf{E} = - \frac{\partial \mathbf{B}}{\partial t}\\ \mathbf{\nabla} \times \mathbf{B} = \mu_0 \epsilon_0 \frac{\partial \mathbf{B}}{\partial t}$$
(Once again, anything that seems incorrect in the above list is intentional.)
[Edited 09/05/2018]
It turned out that my solution was actually complete and it was my supervisor's misunderstanding that more information about the constants were to be obtained. I thank those who shared their ideas.
As you noticed, the first equation is the wave equation, so take the general solution of that for $B$. The solutions you gave were not the most general. And then $$\dfrac{\partial^2 E}{\partial x^2} = -\frac{1}{\mu_0 \epsilon_0} \dfrac{\partial^2 B}{\partial x^2}$$ implies $$ E = - \frac{B}{\mu_0 \epsilon_0} + a(t) x + b(t) $$ where $a$ and $b$ are arbitrary functions of $t$.