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I am trying to solve a vibration problem analytically but I don't understand how to implement the non-homogeneous boundary conditions.

The problem is defined as below:

y[itex]_{t}[/itex][itex]_{t}[/itex](x,t) = a[itex]^{2}[/itex]y[itex]_{x}[/itex][itex]_{x}[/itex](x,t)

With

Boundary conditions:

y(0,t) = 0 [ fixed at zero ]

y[itex]_{x}[/itex](L,t) = [itex]\frac{f(t)}{AE}[/itex] [ Force f(t) at free end x=L ]

Initial condition:

y(x,0) = 0 [ initial displacement = 0 ]

y[itex]_{t}[/itex](x,0) = 0 [ initial velocity = 0 ]

My first question > Is the second BC in it's correct form? I am trying to model a time-dependent force at x=L

second question > How is this problem solved? I tried separation of variables and that didn't work.

any help / resources will be appreciated!

Thanks!