The left hand side of equation1 should look like this, do you agree?
e1L = D[f[et], {et, 3}] + (1 + xi Cot[xi]) f[\[Eta]] D[f[et], {et, 2}] - (D[f[et], et])^2 +
9/8 Sin[2 xi]/xi + \[CapitalLambda] Sin[xi]/
xi (\[Theta][\[Eta]] + N\[Phi][et]) + Subscript[k, 1] (3/2 Sin[xi]/xi - D[f[et], et]) +
Subscript[k, 2] ((3/2 Sin[xi]/xi)^2 - (D[f[et], et])^2)
What is Nphi? And what is f[ et ]? In the right hand side you differentiate f with respect to xi, that should give zero, or I do not understand what you want to do.