Laplace Boundary Value Problem

[lax1113]

Homework Statement

A cantilever beam has uniform load w over a length of L as described by the eq.

EI y'''' = -w y(0) = y'(0) = 0 y''(L) = y'''(L) = 0

EI are constants

find y(x)

Homework Equations

L[y^4] = S^4*Y(s) - S^3*Y(0) - S^2*Y'(0) - s*Y''(0) - Y'''(0)

The Attempt at a Solution

We really didn't do anything similar to this at all in class. All I could think of doing was simply take the transformation of the equation, and we can see that two fo the terms will go to zero, but we are still left with

S^4*Y(s) - s*Y''(0) - Y'''(0) = -w/S

From here what can we really do? I don't see how we can use the boundary conditions because we have Y''(0) not Y''(L) and i don't see how we would ever get to a point that we could use that.

 

[LCKurtz]

You could solve for Y(s) leaving the two boundary conditions in there as unknowns. Then take the inverse transform and use the two conditions in the answer to determine the constants.

But why use LaPlace transforms on this problem anyway? You have an equation of the form

y'''' = constant. Just integrate four times and use the BC's.