To find the Null space of a matrix which contains trig functions. the main objective is to find the green function of the differential equation

Question

(* I am trying to find the null space of a matrix I used NUllspace command but I am getting the empty matrix and I also tried gauss elimination. All I am doing is I am finding out a green function a fourth order differential equation. I Used GreenFunction command in Mathematica I have given 4 boundary condition still I could not able to get the Green function solution *)

        W1 = C2*Sin[\[Beta]*x] + C4*Sinh[\[Beta]*x]  
        W2 = B2*Sin[\[Beta]*(1 - x)] + B4*Sinh[\[Beta]*(1 - x)]  
        (*Compatability condition*)  
        ccd1 = (W1 /. {x -> z}) - (W2 /. {x -> z});  
        ccs1 = ((D[W1, {x}]) /. {x -> z}) - ((D[W2, {x}]) /. {x -> z})  
        ccm1 = ((D[W1, {x, 2}]) /. {x -> z}) - ((D[W2, {x, 2}]) /. {x -> z})  
        ccsh1 = ((D[W1, {x, 3}]) /. {x -> z}) - ((D[W2, {x, 3}]) /. {x -> z})  
        R = Normal@
          CoefficientArrays[{ccd1, ccs1, ccm1, ccsh1}, {C2, C4, B2, B4}][[2]]  
        S = R // MatrixForm  
        NN1 = [NullSpace[R]]  
        (*Greenfunction *)  
        GreenFunction[{(u''''[x] - \[Beta]^4*u[x]), u[0] == 0, u''[0] == 0, 
        u[1] == 0, u''[1] == 0}, u[x], {x, 0, 1}, y]  

Answer 1

As pointed out by aardvark2012, your matrix is invertible and thus its null space is empty. To see for which parameter values it is not invertible, you can do Simplify[Det[S] == 0] to get the simple answer β Sin[β] Sinh[β] == 0. So, for example:

NullSpace[S /. β -> π]

gives

{{Csc[π z] Sin[π (1 - z)], 0, 1, 0}}

and

NullSpace[S /. β -> 8 I π]

gives

{{-((Csch[8 π z] (Cos[8 π (1 - z)] +Cot[8 π z] Sin[8 π (1 - z)]) Sinh[8 π (1 - z)])/(Cosh[8 π (1 - z)] + Coth[8 π z] Sinh[8 π (1 - z)])), 
  Csc[8 π z] Sin[8 π (1 - z)], 
  -((Cos[8 π (1 - z)] Sinh[8 π z] + 
  Cot[8 π z] Sin[8 π (1 - z)] Sinh[8 π z])/(Cosh[8 π z] Sinh[8 π (1-z)] + Cosh[8 π (1 - z)] Sinh[8 π z])), 
  1}}