Consider a laterally loaded long pile in soil. According to the Euler-Bernoulli beam theory, the equation about the displacement may be written as $y''''+xy=0$ where $x\in[0,\infty)$.

One can change this equation into a first order equation by taking the Laplace Transform. Define $Y(s)=\int_0^{\infty}e^{-xs}y(x)dx$.

The equation that $Y(s)$ satisfies is

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