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Mass attached to a vertical spring

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A mass m is attached to a vertical spring (with a spring constant k). Any source of friction is neglected. The equation of motion of the mass is given by

m(d^2x/dt^2) = mg - kx

where g is the constant of gravitation and x refers to the vertical position of the mass. Solve this equation to find the function x(t), assuming that at t = 0 we have x = x_0 and dx/dt = 0. Draw x(t), x(t), and x(t).

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The equation is:
(1.1)

We shift the coordinates to a new coordinate:
...

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