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 $u_{xx}^{\text{'}\text{'}}+y{u}_{yy}^{\text{'}\text{'}}+c\left(y\right)u_y^{\text{'}}-a\left(x\right)u=f\left(x\mathrm{,}y\right)$ in a rectangle leads to problems for the singularly perturbed ordinary differential equation with degeneration $y{Y}^{\text{'}\text{'}}+c\left(y\right)Y^{\text{'}}-({\pi }^2{k}^2+a(y\left)\right)Y=f_k\left(y\right),$$k\in \mathrm{\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 → +∞ and obtain the leading term of the asymptotics in the explicit form.