critical load for buckling

me_ia

1. The problem statement, all variables and given/known data
A rod of leangth L and flexural rigidity EI is pinned at one end by means of torsional spring having a constant beta, and fixed on the other end.
The rod is subjected to a compressive axial force P.
Determine the critical load for instability.(image of the problem is attached)

2. Relevant equations
As far as I understand, the way to solve this problem is:
M(x)=-R*x-P*v(x)+beta*v'(0) , where R is the reaction on the left end at the y direction v(x) is the deflection function and v'(x) is the angle function.
M(x)=v''(x)*IE
3. The attempt at a solution

the two equations above yields:
v''(x)+(P/EI)*v(x)= (beta*v'(0)-R*x)/EI ->

v(x)=A*sin(sqrt(P/EI)*x)+B*cos(sqrt(P/EI)*x)+beta*v'(0)-R*x

to find those constants the boundry conditions are: v(o)=0 v(L)=0 v'(L)=0

How do I represent v'(0) which is unknown ?

Attached Thumbnails