Write the differential equation of motion for a small

Question

The point of suspension of a simple pendulum is given by a harmonic motion x₀ = X_o sin ^wt along a horizontalline, as shown in Figure.

Write the differential equation of motion for a small amplitude of oscillation using the coordinates shown.

Determine the solution for x/x_o, and show that when ω=(√2ω_n )the node is found at the midpoint of I.

Show that in general the distance h from the mass to the node is given by the relation

h=l(ω_n/ω)²,Where ω_n=(√g/l).