pardiffequ = 
        D[z[x, y], {x, 2}] - 5 D[z[x, y],  {x, 1},  {y, 1}] + 
            6 D[z[x, y], {y, 2}] == 0;
sol = DSolve[pardiffequ,  z[x, y],  {x, y}]

r = D[z[x, y], {x, 2}]; 
s =  D[z[x, y],  x,  y]; 
t =  D[z[x, y], {y, 2}];
DSolve[r - 5  s + 6  t == 0,  z[x, y],  {x, y}]

pardiffequ = 
        D[z[x, y], {x, 2}] + 4  D[z[x, y],  {x, 1},  {y, 1}] + 
            4 D[z[x, y], {y, 2}] == 0;
sol = DSolve[pardiffequ,  z[x, y],  {x, y}]

pardiffequ = 
        D[z[x, y], {x, 2}] - 2 D[z[x, y],  {x, 1},  {y, 1}] + 
            5 D[z[x, y], {y, 2}] == 0;
sol = DSolve[pardiffequ,  z[x, y],  {x, y}]

pardiffequ = 
        D[z[x, y],  x] - 4 D[z[x, y],  y] == 
          Sin[x + y];
sol = DSolve[pardiffequ,  z[x, y],  {x, y}]