On the oscillation of certain second-order linear differential equations

This paper consists of three parts: First, letting b(1)(z), b(2)(z), p(1)(z) and p(2)(z) be nonzero polynomials such that p(1)(z) and p(2)(z) have the same degree k >= 1 and distinct leading coefficients 1 and alpha, respectively, we solve entire solutions of the Tumura-Clunie type differential equation f(n)+P(z,f)=b(1)(z)e(p1(z))+b(2)(z)e(p2(z)), where n >= 2 is an integer, P(z,f) is a differential polynomial in f of degree <= n-1 with coefficients having polynomial growth. Second, we study the oscillation of the second-order differential equation f ''-[b1(z)ep1(z)+b(2)(z)e(p2)(z)]f=0 and prove that alpha=[2(m+1)-1]/[2(m+1)] for some integer m >= 0 if this equation admits a nontrivial solution such that lambda(f) s >= 1, we prove that l=2 if the equation f ''-(e(lz)+b(2)e(sz)+b(3))f=0 admits two linearly independent solutions f1 and f2 such that max{lambda(f1),lambda(f2)} Acanthospermum species against herpes simplex virus 1