(a) State a theorem of local existence, uniqueness and $C^{1}$ dependence on the initial data for a solution for an ordinary differential equation. Assuming existence, prove that the solution depends continuously on the initial data.

(b) State a theorem of local existence of a solution for a general quasilinear firstorder partial differential equation with data on a smooth non-characteristic hypersurface. Prove this theorem in the linear case assuming the validity of the theorem in part (a); explain in your proof the importance of the non-characteristic condition.