How to calculate the induced voltage of two electrically isolated coils (solenoids)?
The distance between the coils (solenoids) is so small that can be
  neglected


If that is true then you will have 100% flux coupling and a perfect transformer; Vout will equal Vin if N1=N2. If N1 is 10*N2 then Vout = Vin/10.


  The material used for the coils is copper


Irrelevant except when taking secondary current.


  The coils are wrapped around a plastic core and a laminated core is
  added as displayed here


Largely irrelevant because it is the fundamental premise of the coils being so close that makes flux coupling is 100%. I'm using your words and your definition here - you say the distance can be neglected and when that happens there is perfect coupling period.

However, if the coupling is not perfect then the minefield of math starts with the biot-savart law and would progress onto analyzing the flux density at a distance from the coil: -



From here, to be mathematically rigorous means pages and pages of math and probably some integrations that don't have easy solutions so, if you do want to calculate the flux density for every point inside the target coil (produced by the generator coil) then good luck.