I was given a problem to solve for the speed of a body falling under gravity [equation (1)] where g is acceleration due to gravity, which was easy enough.. but then i thought i would extend it to the case where g is non-constant, and so arrived at equation (2), (where where z is the height above earth [z'=dz/dt and z=dv/dt and z^-2 means z to power -2], and M is the mass of the earth and G is the gravitational constant)
(1) : dv/dt = - g - kv
(2) : z'' + kz' + GMz^-2 = 0
The Attempt at a Solution
I believe this is a non-linear second order DE?? i attempted to solve by setting
z'' + kz' = 0
and solving the complimentary equation, which was OK, but when i came to solve for the particular integral
z'' + kz' = -GMz^-2
i ran into problems, as after substituting in the D and Q operators (http://silmaril.math.sci.qut.edu.au/~gustafso/mab112/topic12/ [Broken]), i could not use the First Shift Theorem, as the RHS is not in an exponential form...
Any ideas anyone?
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