Optimization of the solution to an ODE

Question

Apologies if this is obvious -- I'm very new to Mathematica.

I'm trying to minimize the solution to an ODE with respect to a variable. The following code generates the solution to the ODE,

sol=DSolve[
  {(1/2) * σ^2 * k''[q] + μ*k'[q] - λ*k[q] == 0, 
   k'[0] == -mc, k'[b] == me}, k, q]

but when I try minimizing using the Minimize (with respect to only q) command,

Minimize[k[q] /.First@sol, q]

I'm coming up empty -- it should return the minimum value of k(q)* in terms of $b, \lambda, \mu, \sigma$, $me$ and $mc$ as well as q* in terms of $b, \lambda, \mu, \sigma$, $me$ and $mc$.

* Update *

The following code works (thank you @bbgodfrey):

s = Simplify[k[q] /. DSolve[{(1/2)*σ^2*k''[q] + μ*k'[q] - λ*k[q] == 0,k'[0] == -mc, k'[b] == me}, k[q], q][[1, 1]]]
sq=q /. Solve[D[s, q] == 0, q][[2, 1]]
kq = Simplify[s /. q -> sq]

But a second minimization with respect to b of the function

Gq = kq + b*\[Gamma]

I start to run into trouble again. I try:

Minimize[Gq,b] 

returns unevaluated. This also doesn't work:

sb=b/. Solve[D[Gq, b] == 0, b]

Any help would be very much appreciated!

Answer 1

A slight variant of the code in the question yields

s = Simplify[k[q] /. DSolve[{(1/2)*σ^2*k''[q] + μ*k'[q] - λ*k[q] == 0,
    k'[0] == -mc, k'[b] == me}, k[q], q][[1, 1]]]

(* (E^(-((q (μ + Sqrt[μ^2 + 2 λ σ^2]))/σ^2)) (E^((2 q Sqrt[μ^2 + 2 λ σ^2])/σ^2)
    mc λ σ^2 + E^((2 q Sqrt[μ^2 + 2 λ σ^2] + b (μ + Sqrt[μ^2 + 2 λ σ^2]))/σ^2)
   me λ σ^2 + E^((2 b Sqrt[μ^2 + 2 λ σ^2])/σ^2) mc (μ^2 + λ σ^2 - μ Sqrt[μ^2 + 
   2 λ σ^2]) + E^((b (μ + Sqrt[μ^2 + 2 λ σ^2]))/σ^2) me (μ^2 + λ σ^2 - 
   μ Sqrt[μ^2 + 2 λ σ^2])))/((-1 + E^((2 b Sqrt[μ^2 + 
   2 λ σ^2])/σ^2)) λ (-μ + Sqrt[μ^2 + 2 λ σ^2])) *)

Indeed, Minimize[s, q] returns unevaluated. However, the solution can be obtained by hand, so to speak, by solving D[s, q] == 0 for q.

q /. Solve[D[s, q] == 0, q][[1, 1]]

(* (σ^2 Log[-((E^((b Sqrt[μ^2 + 2 λ σ^2])/(2 σ^2)) 
   Sqrt[E^((b Sqrt[μ^2 + 2 λ σ^2])/σ^2) mc + E^((b μ)/σ^2) me])/
   Sqrt[mc + E^((b μ)/σ^2 + (b Sqrt[μ^2 + 2 λ σ^2])/σ^2)
   me])])/Sqrt[μ^2 + 2 λ σ^2] *)

Presumably, minimization with respect to other parameters can be computed in a similar, although algebraically tedious, manner.