Please any help to solve this problem
Thanks in advance
Our usual approach in helping a student is to find out what the student may have done wrong or misunderstood something. Without showing your work or discussing it we don't know where to focus. HallsofIvy is simply quoting policy, which is pretty much the only way we can focus our attention on what the problem is.
Typically when I do a spring problem I don't worry so much about the negative sign. That makes writing the equations a bit more simple. Just remember that when the spring is extended the spring force is in the direction opposite the extension. (In this case that means we must use the "-" in the spring force. If the spring is compressed, then we use the "+" sign. Of course, this is for a choice of positive direction in the direction that the spring extends.) It's rather like how we choose the direction of the friction in the second part of the problem. The friction force is always acting opposite the direction of the velocity.
So anyway, the equation of motion of our object is
\(\displaystyle \dfrac{d^2x}{dt^2} = - \dfrac{k}{m}x + \dfrac{20}{m}\)
This is the equation from your first attempt.
Your solution method is a bit of overkill. Usually for spring motion we expect some kind of oscillatory motion so it's simpler to just start with the trig functions. So your homogenous solution and particular solution become
\(\displaystyle x_h(t) = A~cos( \omega t) + B~sin( \omega t)\)
\(\displaystyle x_p(t) = C\)
where \(\displaystyle \omega = \sqrt{ \dfrac{k}{m} }\)
Using x(0) = 0 and x'(0) = 0 I get
\(\displaystyle x(t) = \dfrac{20}{k} \left ( cos( \omega t) - 1 \right )\)
just as you got.
I didn't check the rest. If you need further help just let us know.
-Dan