  We construct smooth Riemannian metrics with constant scalar curvature on each
Hirzebruch surface. These metrics respect the complex structures, fiber bundle
structures, and Lie group actions of cohomogeneity one on these manifolds. Our
construction is reduced to an ordinary differential equation called Duffing
equation. An ODE for Bach-flat metrics on Hirzebruch surfaces with large
isometry group is also derived.
