Indeed! Though, it's also true that symbolic manipulations (and exact solutions) give the big allure to CASs

Dan, that is all you are able to say??? This report clearly shows, that Mathematica symbolic capabilities for PDE are dramatically poor. Wolfram developers should work very hard on this topic instead of introducing numerous bizare new functionalities.

u[x, t] := f[x] h[t];
DSolve[{a D[u[x, t], x] + b D[u[x, t], t] == 0}, u[x, t], {x, t}]
    ... DSolve:  f[x] h[t] cannot be used as a function

DSolveValue: "The function h[t] does not have the same number of arguments as independent variables (2)."

and u[x, t] := f[x,0] h[0,t]; ... "f[x,0] h[0,t] cannot be used as a function."

(from links above, a result mathematica "failed")

DSolve[{cw(1,0)(x,t)+w(0,1)(x,t)=0,w(x,0)=f(x),w(0,t)=h(t)},w(x,t),{x,t},Assumptions->c>0 && x>0 && t>0]

https://www.12000.org/my_notes/pde_in_CAS/pdse4.htm#7

I do not think this test is "correct". c>0 does not mean, in Mathematica, that c is a constant.

"therefore w(x(t),t) is constant" (from link above of test results)

olver intro to PDE, transport and traveling wave equations, "in which c is a fixed, nonzero constant" (which i borrowed from the intro (still saving to afford a copy!))

Correct me if i'm wrong that c>0 does not mean "is a constant" in Mathematica but that in Maple it likely does