finding the displacement yxt to the string at any

Question

Finding the displacement y(x,t) to the string at any point at any subsequent time described by a function .

A semi infinite string is initially at rest in a position co-inciting with the position half of the x-axis.

At t=0, the left end of the string begins to more along the y-axis in a manner described by  y(o,t)=  f(x) , where f(x)  is a known function. Find the displacement y(x, t) of the string at any point at any subsequent time. The problem may be stated as follows:

And   y(0,t) = f(x), y(x,t) is bounded as x → ∞ , y(x,0) = 0, y_t(x,0) = 0, special case  f(x) = e^t