A particle of mass 5kg moves along the x axis under the influence of two forces. Firstly, it experiences an attractive force towards the origin O of 40 times the instantaneous distance from O and secondly damping (or resistive force) which is 20 times the instantaneous speed. Assume that the particle starts from 0.2m from the origin.
Set up the differential equation and conditions which describe the motion of the particle.
So What I've tried is letting $a=8s-4v$ where $s$ is displacement, and differentiating to get $v=\frac{e^-{\frac{t+4}4}}4+2s$ Is this the right approach and if so, how can I work out $c$ cos at $t=0$ I don't know $v$, only $s$
from the information provided, you should have $$5a=-40x-20v$$ $$\implies \frac{d^2x}{dt^2}+4\frac{dx}{dt}+8x=0$$
This is easily solved to get a general solution. However with only one initial condition (i.e. only the value of $x$ when $t=0$) you won't be able to get a particular solution.