How to find the stiffness matrix for a beam element?

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I need to find the stiffness matrix of a beam element from the basic priciples. I am not allowed to use Lagrange's polynomial or other methods. Need to find the stiffness matrix from basic principles ie to find at each node k11 and so on..
I have attached a screenshot of the problem.

When i try to find the deflection at each node by using deflection formulae for beams i am not able to find out the answers. the formula i use id EI (d^2 y/ d^2 x)= M , i am not able to find the answers. Please help me out in this
 

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