Asymptotic estimates for the solution of the Cauchy problem for a differential equation with linear degeneration

Application of the method of separation of variables to problems for the linearly degenerate equation $u''_{xx}+yu''_{yy}+c(y)u'_y-a(x)u=f(x,y)$ in a rectangle leads to problems for the singularly perturbed ordinary differential equation with degeneration $yY''+c(y)Y'-(\pi^2k^2+a(y))Y=f_k(y)$, $k\in\mathbb{N}$. In this paper, we examine the asymptotic behavior of solutions of this equation with given initial data at $0$ and zero right-hand side as $k\to+\infty$ and obtain the leading term of the asymptotics in the explicit form.