If all you're doing is showing that a function satisfies the DE, differentiate it to get the derivatives and substitute into the DE. See if it works...
A body of mass 'm' is hung on the end of a vertical spring of spring constant 'k'; the body is pulled down and released. in the resulting harmonic motion, the displacement 'y' of a body is given by the dif eq
m(d^2y)/(dt^2) + ky=0
show that the function:
y= cos(sqrt(k/m))* t
is a solution to the dif eq satisfying the boundary conditions:
y=1 when t=0 and y'=0 when t=0
i see how the differential equation comes from that info, but i'm getting lost in the algebra of trying to simplify its derivatives...