a$)$ A man wants to retire at age $65$ and be able to draw $$10,000$ a year from retirement savings

to supplement his Social Security income and employee retirement benefits. At retirement

he anticipates investing his savings in government securities that should return $5\%$ a year

compounded continuously. By repeating the arguments that led to equation $(2\mathrm{.}8),$ show

that after retirement the principal $P(t)$ in his account satisfies the ODE

$\frac{dP}{dt}=\mathrm{.}05P-10$

where we have used units of $$1000.$

b$)$ Determine the general solution of $(2\mathrm{.}14)$ and find a formula for the solution with, as yet

undetermined, initial value $P_0.$

c$)$ How large must the man's initial retirement savings be so that he can continue drawing his

$$10,000$ income $($annuity$)$ for $20$ years?