Summary: The Dirichlet problem is studied for equations of the form \(0=F(u_{x^ ix^ j},u_{x^ i},u,1,x)\) and also the first boundary value problem for equations of the form \(u_ t=F(u_{x^ ix^ j}u_{x^ i},u,1,t,x),\) where \(F(u_{ij},u_ i,u,\beta,x)\) and \(F(u_{ij},u_ i,u,\beta,t,x)\) are positive homogeneous functions of the first degree in \((u_{ij},u_ i,u,\beta)\), convex upwards in \((u_{ij})\), that satisfy a uniform strict ellipticity condition. Under certain smoothness conditions on F and when the second derivatives of F with respect to \((u_{ij},u_ i,u,x)\) are bounded above, the \(C^{2+\alpha}\) solvability of these problems in smooth domains is proved. In the course of the proof, a priori estimates in \(C^{2+\alpha}\) on the boundary are constructed, and convexity and restrictions on the second derivatives of F are not used in the derivation.