Asymptotics of solutions of a cauchy problem

We consider the Cauchy problem (x)ⁿ=at^r + bx + f(t, x, x), x(0)=0. We prove the existence of continuously differentiable solutions and study their asymptotic behavior for t + 0.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 2, pp. 187–193, February, 1991.