Looks difficult, even setting all constants equal to 1 one gets
In[8]:= DSolve[{D[y[x, t], {t, 2}] + D[y[x, t], {x, 4}] + D[y[x, t], {x, 4}, t] == 0,
(y[x, t] /. x -> 0) == 0, (D[y[x, t], x] /. x -> 0) == 0, (D[y[x, t], {x, 2}] /. x -> L) == 0,
(D[y[x, t], {x, 3}] /. x -> L) == 0}, y[x, t], {x, t}]
Out[8]= (* actually the input *)
so give L a value and try NDSolve:
In[9]:= With[{L = 5},
NDSolve[{D[y[x, t], {t, 2}] + D[y[x, t], {x, 4}] + D[y[x, t], {x, 4}, t] == 0,
(y[x, t] /. x -> 0) == 0, (D[y[x, t], x] /. x -> 0) == 0, (D[y[x, t], {x, 2}] /. x -> L) == 0,
(D[y[x, t], {x, 3}] /. x -> L) == 0}, y[x, t], {x, t}]
]
During evaluation of In[9]:= NDSolve::underdet: There are more dependent variables, {y[x,t],(y^(0,1))[x,t],(y^(0,2))[x,t]}, than equations, so the system is underdetermined. >>
Out[9]= (* actually the input *)
first job you should work on is to get a numerical solution with unit constants. Seemingly the problem is incorrect formulated.