$12$ A financial analyst predicts that the price of shares in a company $($a financial product$)$ can be modelled by

$\frac{d^{2}p}{(d)t^2}+1\mathrm{.}5\frac{dp}{(d)t}+5p=10-e^{-t}$

where $p$ is the price in dollars and $t$ is the time in years.

The initial price is $$1$ and it is expected to initially fall at a rate of $$2$ per year. Use Euler's method with a step length of $0\mathrm{.}1$ to determine tor $1$ decimal place, according the model,

a when the analyst should purchase the shares

b how long the analyst should keep the shares

c how much profit the analyst will make per share

d the long$-$term price of the shares.

e By considering the observed behaviour, suggest why the model is unlikely to be accurate.