where parameters a,b,c,e are real nonzero constants; is a complex constant; function is a function mapping from real number y to a complex number. The goal is to solve for function . This is what I have done. Solve this differential equation by integrating with respect to y:
where is another complex constant. Plugging in y=0 and using the fact that g(0)=0, we have . Therefore, we have
The background of this problem is Cauchy functional equation, so my conjecture is one solution could be . Plugging in , I get , which implies that one solution is . Then, I move on to show uniqueness. I define a vector-valued function such that
where . Then, I rewrite this differential equation as
where . By the uniqueness theorem of first order differential equation, solution h(y) is unique. I have two questions. First, I think equation (1) and (2) should be equivalent. However, it seems that equation (1) can imply equation (2) but equation (2) may not imply equation (1). This is because h(0)=0 may imply either or . Second, I have only proved that h(y) is unique. How should I proceed to show g(y) is also unique.