You can use the coeffs function to equate like powers of t and obtain the corresponding ODEs.
eqn = lhs(EQ12)-rhs(EQ12) == 0
c = coeffs(eqn,t)
Here, c is a 1*6 symbol in which c(1,1) = "- diff(F(z), z, z) + diff(F(z), z) + F(z)^2" which is the coefficient of t. Similarly, you have coefficients for other powers of t. Now, you can use the equation c(1,1) == 0 for further calculations.
For more information, refer the following links