How to solve this ODE by DSolve or NDSolve?

Question

I try to solve this second order ODE to get k:

$ \frac { \partial }{\partial z} ~ \frac{ \partial } {\partial \bar{z}}~ k[z, \bar{z}] = 5 $

Where z is complex coordinates, so to take the derivative for it, I have used ComplexD function defined in this thread:

What is the best way to define Wirtinger derivatives

So that the solution I have tried:

(* First: take the derivative *)

D[ ComplexD[k[Conjugate[z]], Conjugate[z]], z]

(* the output gives *)

Conjugate’[z] k’’[Conjugate[z]]

(* Second to solve I called Conjugate[z] by x *)

DSolve[ x’ k’’[x] == 5 , k[x], x]

Which gives:

The question :

• Are these steps right? Sure there’s a better way for solution, but I don’t want to use NIntegrate.

• What does & mean here ( i know it means pure function, but then what’s the value of k[x] ?

• How to determine the value of k[x] for specific values of C[1] and C[2] ?

Answer 1

In DSolve[ x' k''[x] == 5 , k[x], x] Mathematica interprets x' incorrectly.

Since $ \dfrac{\mathrm{d}[f(x)^*]}{\mathrm{d}x} = \biggl[\frac{\mathrm{d}f(x)}{\mathrm{d}x}\biggr]^* $ you can drop x' from your equation. Then

{{sol}}=DSolve[ k''[x] == 5 , k[x], x]

gives you

{{k[x] -> (5 x^2)/2 + C[1] + x C[2]}}

which does provide you with the solution of your differential equation that you can use like this:

sol /. {C[1] -> 1, C[2] -> 3}

(* k[x] -> 1 + 3 x + (5 x^2)/2 *)

or like this

ksol = Function[{x}, Evaluate[k[x] /. sol /. {C[1] -> 1, C[2] -> 3}]]

(* Function[{x}, 1 + 3 x + (5 x^2)/2] *)

ksol[5]

(* 157/2 *)

or like this

ksol = Function[{x, c1, c2}, 
  Evaluate[k[x] /. sol /. {C[1] -> c1, C[2] -> c2}]]

(* Function[{x, c1, c2}, c1 + c2 x + (5 x^2)/2] *)

ksol[5, 1, 3]

(* 157/2 *)