Second Order Homogeneous Equation IVP

I am stuck on this question, I have done a little but I am generally lost. The second order equation is the general form of modeling springs (g is for gamma, but g is easier to type)

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Show that the solution of the initial value problem

my'' + gy' + ky = 0, y(t0) = y0, y'(t0) = y1

can be expressed as the sum y = v + w, where v satisfies the initial conditions

v(t0) = y0, v'(t0) = 0

w satisfies the initial conditions

w(t0) = 0, w'(t0) = y1

and both v and w satisfy the same differential equation as u.

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What I have done is solved the auxiliary equation mλ² + gλ + λ = 0

Giving, λ = (g/2m)(-1 ± (1 - (4km/g²))^0.5)

So there are three possible cases:

1: If g² - 4mk > 0

y = A*exp(λ1 t) + B*exp(λ2 t)

2: If g² - 4mk = 0

y = (A + Bt)*exp(-gt/2m)

3: if g² - 4mk < 0

y = exp(-g/2m)*(Acos(vt) + Bsin(vt))

I really don't know if i'm on the right track, any help would be much appreciated! (Talking)