\(\(SetDirectory["c:\myDirectory"];\)

      << recursionOperator.m\)

RecursionOperator[
D[u[x, t], t]  ==  
        6*u[x, t]*D[u[x, t], x] + D[u[x, t], {x, 3}], 
u[x, 
        t], 
{x, t}
]

RecursionOperator[
D[u[x, t], t]  ==  
        3*D[u[x, t], x]^2 + D[u[x, t], {x, 3}], u[x, t], {x, t}]

RecursionOperator[
D[u[x, t], t]  ==  
        u[x, t]^2*D[u[x, t], x] + D[u[x, t], {x, 3}], 
u[x, 
        t], {x, t}]

RecursionOperator[
      D[u[x, t],  t]  == u[x, t]*D[u[x, t], x] + D[u[x, t], {x, 2}], 
      u[x, t], {x, t}]

RecursionOperator[
D[u[x, t], 
          t] == - u[x, t]*D[u[x, t], x]  +  ν*
            D[u[x, t],  {x, 2}], 
u[x, 
        t], 
{x, 
        t}, 
{ν}
]

\(RecursionOperator[
      D[u[x, t], \ t]\  == \ 
        20*u[x, t]^2*D[u[x, t], x] + 25*D[u[x, t], x]*D[u[x, t], {x, 2}] + 
          10*u[x, t]*D[u[x, t], {x, 3}] + D[u[x, t], {x, 5}], 
      u[x, t], {x, t}, \ Seed -> 2
]\)

\(RecursionOperator[
      D[u[x, t], \ t]\  == \ 
        5*u[x, t]^2*D[u[x, t], x] + 25/2*D[u[x, t], x]*D[u[x, t], {x, 2}] + 
          5*u[x, t]*D[u[x, t], {x, 3}] + D[u[x, t], {x, 5}], 
      u[x, t], {x, t}, \ Seed\  -> \ 2
]\)

\(RecursionOperator[
      D[u[x, t], \ t]\  == \ 
        D[u[x, t], \ {x, 5}]\  + \ 
          10*D[u[x, t], \ x]*D[u[x, t], \ {x, \ 3}]\  + \ 
          15/2*D[u[x, t], \ {x, \ 2}]^2\  + \ 
          20/3*D[u[x, t], \ x]^3, 
u[x, t], {x, t}, 
      Seed\  -> 2
]\)

RecursionOperator[
      D[u[x, t],  t]  ==  
        5*u[x, t]^2*D[u[x, t], x] + 5*u[x, t]*D[u[x, t], {x, 3}] + 
          5*D[u[x, t], x]*D[u[x, t], {x, 2}] + D[u[x, t], {x, 5}], 
      u[x, t], {x, t},  Seed  ->  2]

\(RecursionOperator[
      D[u[x, t], \ t]\  == \ 
        5/3*D[u[x, t], x]^3 + 5*D[u[x, t], x]*D[u[x, t], {x, 3}] + 
          D[u[x, t], {x, 5}], u[x, t], {x, t}, \ 
      Seed\  -> \ 2
]\)

RecursionOperator[
{D[u[x, t], t]  ==  
          1/2*(6*u[x, t]*D[u[x, t],  x]  +  D[u[x, t],  {x, 3}]) - 
            2*v[x, t]*D[v[x, t],  x], 
D[v[x, t],  
            t]  ==  -3*u[x, t]*D[v[x, t],  x]  -  
            D[v[x, t],  {x, 3}]}, 
{u[x, t], v[x, t]}, {x, 
        t},  Seed -> 2
]

RecursionOperator[{D[u[x, t],  t]  ==  
          D[u[x, t], {x, 2}] + v[x, t]^2, 
        D[v[x, t], t]  ==  D[v[x, t], {x, 2}]}, {u[x, t], 
        v[x, t]}, {x, 
        t}, 
WeightRules  ->  {Weight[u]  ->  
            Weight[v]}]

RecursionOperator[{D[u[x, t],  t]  ==  
          u[x, t]*D[v[x, t], x] + D[u[x, t], x]*v[x, t], 
        D[v[x, t],  t]  ==  
          D[u[x, t], x] + v[x, t]*D[v[x, t], x]}, {u[x, t], v[x, t]}, {x, 
        t},  WeightRules  ->  {Weight[u]  ->  2}]

RecursionOperator[{D[u[x, t],  t]  ==  
          2*u[x, t]^2*v[x, t] - D[u[x, t], {x, 2}], 
        D[v[x, t],  t]  ==  -2*u[x, t]*v[x, t]^2 + 
            D[v[x, t], {x, 2}]}, {u[x, t], v[x, t]}, {x, t},  
      WeightRules -> {Weight[u]  ->  Weight[v]}]

RecursionOperator[
      D[u[x, t],  t]  ==  
        30*u[x, t]^2*D[u[x, t], x] + 20*D[u[x, t], x]*D[u[x, t], {x, 2}] + 
          10*u[x, t]*D[u[x, t], {x, 3}] + D[u[x, t], {x, 5}], 
      u[x, t], {x, t}]