For the beam shown below, write the moment equation at the section plane and then using the double integration method, determine the location along the beam where maximum deflection occurs and then calculate the maximum deflection.

$EI={\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}Mdx$

EIy$=Mdx$

Where: $E=$ Modulus of elasticity;

I $=$ second moment of area of the beam;

$=$ gradient of the beam;

$y=$ the deflection distance;

$M=$ the moment equation for the load case on the beam.