Solving Klein Gordon’s equation

My solution:

Let ## \phi (x, t) = F(x) A(t) ##, then Eq. (1) becomes

##
\frac{1}{A(t)} \frac{\partial^2}{\partial t^2} - \frac{1}{F(x)} \frac{\partial^2}{\partial x^2} = 0
##

So that : ## \frac{\partial^2}{\partial t^2} = k^2 ~A (t)##, and ## \frac{\partial^2}{\partial x^2} = k^2 ~F (x)##.

Leads to :
##
\phi(t,x) = ( c_1 e^{kt} + c_2 e^{-kt} ) ( c_3 e^{kx} + c_4 e^{-kx} )
##

Assuming BC and IC :

##
bc={\phi[t,0]==1,(D[\phi[t,x],x]/.x->Pi)==0}
##
##
ic={\phi[0,x]==0,(D[\phi[t,x],t]/.t->0)==1}
##

BC leads to ##c_3 = c_4= 1/2 ## and the IC leads to to ##c_1=- c_2= 1/2 ##.

Ending up by :
##
\phi(t,x) = \frac{1}{4} e^{k(t-x)} - \frac{1}{4} e^{-k(t-x)}……… (2)
##

So any help are these steps correct till Eq. (2) ? And how to determine ##k##?

You just wrote down the scalar wave equation, not the Klein-Gordon equation.