Finding a solution to Maxwell's equations from initial datas

[fluidistic]

Homework Statement

Suppose we know that $$B(\vec x ,t)$$ is a solution to Maxwell's equations in vacuum and furthermore we know that $$E(\vec x , 0)=E_0$$.
How do we find $$E(\vec x , t)$$?

Homework Equations

$$\nabla \cdot E = 0$$.
$$\nabla \cdot B =0$$.
$$\vec \nabla \times \vec B = \left ( \frac{-1}{c} \right ) \cdot \frac{\partial E}{\partial t}$$
$$\vec \nabla \times \vec E = \left ( \frac{1}{c} \right ) \cdot \frac{\partial B}{\partial t}$$.
I'm using Gaussian's units.

The Attempt at a Solution

I think I could work with the 2 lasts equations I posted to find E but I don't reach anything. I'd like a very small guidance like if I'm in the right direction + a hint if possible.
Thanks.

 

[nickjer]

Think about integrating the 3rd equation with respect to time.

 

[fluidistic]

Think about integrating the 3rd equation with respect to time.

Thanks for the tip.
I reach $$\vec E=-c \int \vec \nabla \times \vec B dt$$. I think of using the initial condition. So $$E_0=-c\int \vec \nabla \times \vec B (\vec x ,0) dt$$.
I'm stuck here.

 

[nickjer]

The integral will have bounds.