Let z=u', then z'=u''=-exp(x)*u.
So the planar system is {u'=z,z'=-exp(x)*u}, or in matrix form, U'=A*U, where U is the column vector (u,z) and A is the 2x2 matrix (0,1,-exp(x),0).
What can be said about the Poincare return map on the transversal u>0?