pde equation complex plane

I'm trying to solve the following differential equation:

DSolve[{D[χ[β, βc, t],t] == -Γ (β D[χ[β, βc,t], β] + βc D[χ[β, βc, t], βc])}, χ, {β, βc, t}]

and this is the output

{* {χ -> Function[{β, βc, t}, C[βc/β, (t Γ - Log[β])/Γ]]} *}

Now:

1. I don't understand why the result is written as a function of two arguments
2. What is the meaning of (t Γ - Log[β])/Γ?
• First, the solution is not a function of two arguments but of three {\[Beta], \[Beta]c, t}. Your solution is given in the form of PureFunction. For details about pure function see this link. Regarding last part of your question, t[CapitalGamma]-Log[[Beta]])/[CapitalGamma] is part of the solution (again see link for pure function). – ercegovac Feb 6 '17 at 19:15

eqn = D[χ[β, βc, t],
t] == -Γ (β D[χ[β, βc,
t], β] + βc D[χ[β, βc, t], βc]);

soln = DSolve[eqn, χ, {β, βc, t}][]

{χ -> Function[{β, βc, t},
C[βc/β, (t Γ - Log[β])/Γ]]}

Verifying the solution

eqn /. soln // Simplify

(*  True  *)

C is an arbitrary function of its two shown parameters. For example. let C be the undefined function f

soln2 = soln /. C -> f

(*  {χ -> Function[{β, βc, t},
f[βc/β, (t Γ - Log[β])/Γ]]}  *)

As expected, this solution also satisfies the equation

eqn /. soln2 // Simplify

(*  True  *)
• I tried to impose χ[β, βc, 0]==E^(-(β βc)/2 but i didn't get anything – Kowalski Feb 6 '17 at 20:50
• @Kowalski - your condition appears to be inconsistent with the general solution. \[Chi][\[Beta], \[Beta]c, 0] must be expressible as a function of the arguments {\[Beta]c / \[Beta], -Log[\[Beta]] / \[CapitalGamma]} – Bob Hanlon Feb 6 '17 at 23:59
• But I found that also [Chi] -> Function[{β, βc, t}, C[β E^(-Γt), βc E^(-Γt)]], is a solution, so my condition is not inconsistent, i'm right? – Kowalski Feb 7 '17 at 8:28
• @Kowalski - I'm not a mathematician and I don't know how to force Mathematica to provide a result using your constraint. – Bob Hanlon Feb 7 '17 at 16:26