Substitute directly the U(z) into the differential equation.

Then assume d is an integer and compare the coefficients of e^bz, e^bnz and the constant term, it turns out that d=2, b=+-1/root6, c=1/root2 and a is arbitrary.

Then assume d is an integer and compare the coefficients of e^bz, e^bnz and the constant term, it turns out that d=2, b=+-1/root6, c=1/root2 and a is arbitrary.