Why can’t DSolve handle `df/dt = d^2f/dx^2`?

The output of DSolve[D[f[x, t], t] == D[f[x,t],{x, 2}], f[x,t], {x,t}] is just a copy of the input — does this mean DSolve couldn’t find an answer?

There are fairly trivial solutions, e.g. f(x,t) = e^(x+t)