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The following code solves your equation numerically (with 4th order Runge-Kutta method) for y(0)=0.5 and plots the solution:

x,y=var('x,y')
mynumericalsolution=desolve_rk4(tan(y) * (y^3 - y), dvar=y, ivar=x, ics=[0,0.5],end_points=10, step=0.1)
myplot=line(mynumericalsolution,axes_labels=[x,y])
myplot.show()

I will edit if I can find an exact solution with SageMath.

The following code solves your equation numerically (with 4th order Runge-Kutta method) for y(0)=0.5 and plots the solution:

x,y=var('x,y')
mynumericalsolution=desolve_rk4(tan(y) * (y^3 - y), dvar=y, ivar=x, ics=[0,0.5],end_points=10, step=0.1)
myplot=line(mynumericalsolution,axes_labels=[x,y])
myplot.show()

I will edit if I can find an exact solution with SageMath.think SageMath cannot solve it symbolically.