Can I use GeneratedParameters -> (Module[{C}, C] &) so that every time I execute DSolve achieve the continuous integration constants?

In my case is the partial derivative of u[x,t]. This is what I have written:

constantCounter = 1;
DSolve[ D[u[x, t], {x, 2}] == -(3 /(2*k*\[Theta])) x  D[u[x, t], t] - 
   k  3 /(2*k*\[Theta]) x  (\[Theta] - x) D[u[x, t], x] + 
   3 /(2*k*\[Theta])  x*x * 
    u[x, t] (*y''[x]==y[x]*)(*EDs[[5]]*), u, x, t, 
 GeneratedParameters -> (C[constantCounter++] &)]

This is the Output:

DSolve[
\!\(\*SuperscriptBox[\(u\), 
TagBox[
RowBox[{"(", 
RowBox[{"2", ",", "0"}], ")"}],
Derivative],
MultilineFunction->None]\)[x, t] == (3 x^2 u[x, t])/(2 k \[Theta]) - (
   3 x 
\!\(\*SuperscriptBox[\(u\), 
TagBox[
RowBox[{"(", 
RowBox[{"0", ",", "1"}], ")"}],
Derivative],
MultilineFunction->None]\)[x, t])/(2 k \[Theta]) - (
   3 x (-x + \[Theta]) 
\!\(\*SuperscriptBox[\(u\), 
TagBox[
RowBox[{"(", 
RowBox[{"1", ",", "0"}], ")"}],
Derivative],
MultilineFunction->None]\)[x, t])/(2 \[Theta]), u, x, t, 
 GeneratedParameters -> (C[constantCounter++] &)]

No evaluation. Please advise how to enter partial differential equation for this GeneratedParameters command to work.

This is what I've written in place of the equation. I only replaced the differential equation.

'eq = D[u[x, t], {x, 2}] == -(3 /(2*k*\[Theta])) x  D[u[x, t], t] - 
    k  3 /(2*k*\[Theta]) x  (\[Theta] - x) D[u[x, t], x] + 
    3 /(2*k*\[Theta])  x*x * u[x, t];
sol = DSolve(*Value*)[eq, u, {x, t}, 
  GeneratedParameters -> (C[constantCounter++] &)]
eq /. u -> sol /.
 {t -> 0, x -> 1, C[1 | 2] -> 0, C[4 | 5] -> 1, C[7 | 8] -> 0, 
  C[6] -> 1}'

However, there are errors:

• DSolve::dsvar: 1 cannot be used as a variable.
• General::stop: Further output of DSolve::dsvar will be suppressed during this calculation.
• Increment::rvalue: constantCounter is not a variable with a value, so its value cannot be changed.