| Popis: |
This thesis starts from the following observation; if v;w are solutions of y" + Py = 0 where P is entire, and v and w are both 0 at z0 ε C, then W(v,w) = vw' - v'w ≡ 0 and v,w are linearly dependent. It is then natural to ask what happens if v;w solve different equations, but have (mostly) the same zeros. The case where the first equation is of the second order and has a polynomial coefficient while the second equation is of order greater than one with entire coefficients was investigated first, and some relations between the solutions and between the coefficients were proved. We next obtained approximately the same results when a transcendental coefficient was considered instead of a polynomial in the first equation, but with some amendments to the conditions. We then examined the case where the equations are non-homogeneous of the first order and determined what the solutions have to be. We also could determine the solutions in the case where the equations are a combination of homogeneous and non-homogeneous equations. Finally, the case where the solutions take the value 0 and a non-zero value at mostly the same points was studied, and again the solutions were determined. In order to prove our results, we used some background from Nevanlinna theory and some of its applications. |
