Consider an infinitely long beam on an elastic foundation $($see sketch in class notes$).$ Let $k$ be the

"foundation modulus" $($i$.$e$.,$ spring stiffness per unit $x$ length$)$ and $w(x)$ is a prescribed loading $($force

per unit length$).$ Physically, the beam might be a train track, with the elasti foundation used to model

the track bed. According to the Eurler beam theory, the vertical deflection $u(x)$ resulting from the

load distribution due to the applied load and elastic foundation is given by

$EI{u}^{\text{'}\text{'}\text{'}\text{'}}=w(x)-ku,$

where $EI$ is the flexure rigidity of the beam $(E$ young's modulus and I cross$-$section moment of area$).$

Here, we neglected the mass of the beam. Solve for the deflection $u(x)$ of the beam when

$w(x)$ is $4a-$periodic square wave $w(x)=w_o$ non$-$zero constant for $w(x)=0w(x)-.a.$

$w(x)is$ not periodic and has infinite support $-.-a,$ and $w(x)=0$ for

$a.$

$w(x)is$ not periodic and has infinite support $-.$