Example Notebook, Mechanics

A particle is under the influence of a force F=-kx+((kx^3)/(α^2)), where k and α are constants and k is positive. Determine U(x) and discuss the motion.

Manipulate[
Plot[
{(1/2)k (x^2)-(1/4)((k (x^4))/(α^2)),(1/4)k (α^2)},{x,-b,b},
PlotRange->{{-b,b},{-25,a}},
Frame->True,
FrameLabel->{"x","U"},
GridLines->{Range[-b,b,2],Range[-20,a,10]},
PlotStyle->{{Thick,Red},{Dashed,Blue}}
],
{{α,6},0.1,10,0.01, Appearance->"Labeled"},
{{k,2},0,4,0.01, Appearance->"Labeled"},
{{b,10},1,50,1, Appearance->"Labeled"},
{{a,50},25,100,1, Appearance->"Labeled"}
]