Asymptotics of the solution of a system of nonlinear differential equations with a small parameter and with a higher-order extinction effect

We consider the Cauchy problem for two nonlinear differential equations with a small parameter multiplying the derivative in one of the equations. The right-hand side of this equation has a zero of high order at the origin with respect to one of the unknown functions. We construct and justify a uniform asymptotic approximation to the solution with accuracy of any power of the small parameter. We reveal two boundary layers in a neighborhood of the initial point.