If the following equation establishes equilibrium in slightly bent position, the mid-span deflection of a member shown in the figure is
$$\frac{d^2y}{dx^2}+\frac{P}{EI}y=0$$
If $a$ is amplitude constant for $y,$ then
- $y=\dfrac{1}{P} \bigg( 1- a \cos \dfrac{2 \pi x}{L} \bigg) \\$
- $y=\dfrac{1}{P} \bigg( 1- a \sin \dfrac{2 \pi x}{L} \bigg) \\$
- $y= a \sin \dfrac{n \pi x}{L} \\$
- $y= a \cos \dfrac{n \pi x}{L}$