Let c be a positive number. a differential equation of the form

dy
dt= ky1 + c
where k is a positive constant, is called a doomsday equation because the exponent in the expression ky1 + c is larger than the exponent 1 for natural growth.

(a) determine the solution that satisfies the initial condition

y(0) = y0.

ans: y=(-ckt+y0^-c)^(-1/c)

(b) show that there is a finite time
t = t
(doomsday) such that
lim t %u2192 t%u2212y(t) = infinity.
ans: y(t) --> infinity as 1-cy0^ckt--> 0 as t--> 1/cy0^ck
t=1/cy0^ck, them lim y(t)=infinity as t--> t-