# Green's Function with repulsive force

1. Sep 30, 2015

### teme92

1. The problem statement, all variables and given/known data
Consider an object subject to a linear repulsive force, $F = kx$. Show that the Green’s function for this object is given by:

$G(t-t^\prime)=\frac{1}{m\gamma}sinh(\gamma(t-t^\prime))$

where $\gamma=\sqrt{\frac{k}{m}}$

2. Relevant equations
$sinhx=\frac{e^x+e^{-x}}{2}$

3. The attempt at a solution
So when I'm trying to make the Green's Function I start with:

$x(t)=\int_0^tG(t-t^\prime)F(t^\prime)dt^\prime$

$F(t^\prime)=kx$ so:

$x(t)=\int_0^tG(t-t^\prime)(kx)dt^\prime$

For an impulse I think $G(t-t^\prime)=\frac{v(t-t^\prime)}{\Delta p}$ where $\Delta p = mv-0=mv$. However I don't know what to do for a repulsive force. Also I don't know how to get to the $sinh$ part. Any help would be greatly appreciated.

2. Sep 30, 2015

### gabbagabbahey

Hint: $F=kx$ tells you that $m\ddot{x}(t) = kx(t)$...what does your equation for the Green's Function tell you $\ddot{x}(t)$ is?