Example 3: Buckling of columns (Euler-Bernoulli Equations)

The lateral displacement at position of a clamped-hinged bar of length that is subjected to a load may be approximated by the Euler-Bernoulli equations

In the boundary conditions

The parameter , where is the flexural rigidity of the bar.

This is an example of a Boundary Value Problem (BVP) for a fourth order linear ODE.

Note: Observe that is a solution of this problem in all value of and . A more interesting problem is to determine and those values of and for which non-trivial solutions exist. As such, this BVP is also a differential eigen value problem. Non- trivial solutions are called eigen functions and their corresponding values of are eigen values.