sage: diff(u, x, y) + 4 * diff(u, x)
-2*x*(e^(-4*y) - 1) + 2*x*e^(-4*y)

sage: u.subs(y = x^2)
1/3*x^6 + x^4 - 1/3*(x^5 + 3*x^3 - 3*x - 3)*x

Months later of my original question,

there is the simplify() method which usually doesn't do much (or did not in my experience); but there's also full_simplify(), and the latter might do some real work on an expression!