On factorizable solutions of the differential equation \((f')^2=a_0(z)(f-a_1(z))^2f\) (Q1387441)

The authors consider the differential equation \[ (f')^2= a_0(z)(f- a_1(z))^2f \] and show that in all possible nontrivial factorization \(f(z)= h(g(z))\), the growth of the right factor \(g(z)\) can be controlled by the growth of the coefficients. This result has been proven earlier by others but with extra conditions imposed. Here it is shown that the extra conditions can be eliminated. The following two results are given: (1) If \(f(z)= h(g(z))\), where \(g(z)\) is transcendental entire and \(h\) is a nonconstant meromorphic function then \(T(r, g)= O(H(r))\), and (2) If \(f(z)= h(g(z))\) is the solution to the differential equation, with \(a_1(z)\) not identically zero, \(h\) is as before and \(g\) satisfies \(H(r)= O(T(r, g))\), then \(\lambda(h)= \rho(h)={n\over 2}\), \(n\in\mathbb{Z}^+\).