Chapter 2.4, Problem 4SA

Add a sinusoidal pile to the beam. This means adding a function of form $s(x)=-pg{\sin }_{\frac{x}{L}}x$ to the force term $f(x)$ . Prove that the solution $y\left(x\right)=\frac{f}{24EI}{x}^2\left(x^2-4Lx+6Lx\right)-\frac{pgL}{EI\pi }\left(\frac{L^3}{{\pi }^3}\sin \frac{\pi }{L}x-\frac{x^3}{6}+\frac{L}{2}{x}^2-\frac{L^2}{{\pi }^2}x\right)$ satisfies the Euler-Bernoulli beam equation and the clamped-free boundary conditions.