Only the p=0.5 is correct how about the others?

One possible problem with your solution is that it contains in the argument of cosine, when it should be a linear term. Aside from that, the best way to track down a mistake is to start from the beginning:

The mass's position function satisfies the second-order ODE



(assuming there are no other external forces acting on the mass). The characteristic equation for this ODE is



which means the general solution to this ODE is



The angle difference identity for cosine allows you to condense the trigonometric part of the solution to



where and , leaving you with



These unknown constants can be found explicitly, as and .

Given that and , and the solution's first derivative is



you have the following system of equations needed to find , and from there the corresponding values of and .



So the particular solution is




In terms of what you should submit, you would use






or rely on the exact forms in case rounded answers are not accepted.

The answer of the above question is

The correct answer is c