<< Calculus`DSolveIntegrals`

DSolve[
      (KbT/ζ)*g''[x] + 
          (kf/ζ)*
            Exp[-k*x]*g'[
              x] + (-k*(kf/ζ)*Exp[-k*x] + (Subscript[ρ, m])^2)*g[x] == 
        0, g[x], x]

DSolve[
      D*z''[x] + 
          Exp[-k*x]*z'[x] + (-k*Exp[-k*x] + c^2)*z[x] == 
        0, z[x], x]

DSolve[
      D*z''[x] + 
          Exp[-k*x]*z'[x] + (-k*Exp[-k*x] + c^2)*z[x] == 
        0, z[x], x]

( * O - 
            U process (separated into eigen functions)  is also solved in terms of 1  F1 *)

DSolve[y''[x] + k*x*y'[x] + (k + c^2)*y[x] == 0, y[x], 
      x]

DSolve[
      D[p[x, t], t] - k*p[x, t] - D[p[x, t], x] - T*D[p[x, t], x, x] == 0, 
      p[x, t], {x, t}]

DSolve[D[p[x, t], t] - D[p[x, t], x] - T^2*D[p[x, t], x, x] == 0, 
      p[x, t], {x, t}]

\(CompleteIntegral[
\ \ \ \ 
      \(\(Derivative[0, \ 1]\)[u]\)[x, \ y]\  == \ 
        \((u[x, \ y]\  + 
\ \ \ \ \ \ \ \ 
              x^2*\(\(Derivative[1, \ 0]\)[u]\)[x, \ y]^2)\)/y, 

      \ \ \ \ \ \ \ \ u[x, y], \ {x, y}]\)

Xav = Xo*Exp[-k*t]\(σ2 = 	D/\((2*k)\)*\((1 - Exp[\(-2\)*k*t])\)\)

\(\(
p = 
      \((1/\@\(2*Pi*σ2\))\)*
        Exp[\(-\((x - Xav)\)\)/\((2*σ2)\)]\)\)

D[p, x]

D[p, t]

D[p, x, x]

Rhs = D[p*x]

k =.

<< Calculus`DSolveIntegrals`

DSolve[z''[x] + 
          Exp[-k*x^2]*z'[x] + (-k*2*x*Exp[-k*x^2] + c^2)*
            z[x] == 0, z[x], x]

DSolve[
      z''[x] + k/(k + x^2)*z'[x] + 
          (-k*2*x/((k + x^2)^2) + c^2)*z[x] == 0, z[x], x]

DSolve[z''[x] + 
          Exp[-k*Abs[x]]*z'[
              x] + (-k*Abs[x]*Exp[-k*Abs[x]] + c^2)*z[x] == 0, 
      z[x], x]