Suppose a uniform beam of length is simply supported at and . If the load per unit length of the beam is given by w(x), , then the differential equation modeling the deflection of the beam y(x) is given by Because the beam is simply supported at both ends we know that

We are going to use a Fourier series to solve this equation. To satisfy the initial conditions we should use a expansion of :

Plugging the last sum into the differential equation we get that

Using a Fourier Sine series expansion on we know that


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