determining displacement of columns equation coefficients

[psuaero]

1. The problem statement, all variables and given/known data
Determine the constants c_{1},c_{2},c_{3},c_{4}

of the column displacement equation where k=\sqrt{\frac{P}{EI}}

In lab we subjected 3 specimens to a compressive load in a pinned-pinned and clamped clamped configurations. I have to compare theory results to experimental.


2. Relevant equations
displacement: w(x)=c_{1}sin(kx)+c_{2}cos(kx)+c_{3}x+c_{4}


3. The attempt at a solution

I know i have the following boundary conditions for pinned pinned:
w(0)=0 , EI*w''(0)=0 , w(L)=0 , EI*w''(L)=0

and for clamped clamped
w(0)=0 , w'(0)=0 , w(L)=0 , w'(L)=0

I found that
w'(x)=kc_1cos(kx)-kc_2sin(kx)+c_3

w''(x)=-k^2c_1sin(kx)-k^2c_2cos(kx)

using boundary conditions for simply supported I arrive at system of equations:
w(0)\rightarrow c_2+c_4=0

w(L)\rightarrow c_1sin(kL)+c_2cos(kL)+c_3L+c_4=0

w''(0) \rightarrow -k^2c_2=0]

w''(L)\rightarrow -k^2c_1sin(kL)-k^2c_2cos(kL)=0

using clamped clamped boundary conditions
w(0)\rightarrow c_2+c_4=0

w'(0)\rightarrow c_1k+c_3=0

w(L)\rightarrow c_1sin(kL)+c_2cos(kL)+c_3L+c_4=0

w'(L)\rightarrow c_1kcos(kL)-c_2ksin(kL)+c_3=0

I tried putting the above in matrix form and solve simultaneously but only achieved the trivial solution. I was thinking of finding the determinate of the matrices and plotting them but not sure if that would provide the correct solution. any suggestions?
1. The problem statement, all variables and given/known data



2. Relevant equations



3. The attempt at a solution