A research note on the second order differential equation

Let U(t,λ) be a solution of the Dirichlet problem y"+(λt-q(t))y=0 -1⩽t⩽1 y(-1)=0=y(x), with variable t on (-1,x), for fixed x, which satisfies the initial condition U(-1,λ)=0, ∂U/∂t(-1,λ)=1. In this paper, the asymptotic representation of the corresponding eigenfunctions of the eigenvalues has been investigated. Furthermore, the leading term of the asymptotic formula for ∂U/∂λ(x,λ_n(x)),λ\´_n (x) and ∫_-1^x (υ,λ_n)dυ is obtained where λ_n (x) is a negative eigenvalue of the Dirichlet problem on [-1,x] with fixed x<0