Problem 1. (2 marks)

the c(S)o(h)mm(ow t)o(h)n i(at)nt(ts)er(i¯n)v(−)al(1t)f(+)or(√)wh(1 −)i(t)c(2)h(is)th(a)e(s)s(o)ol(lu)u(ti)ti(o)o(n)n(o)and(f the) d(d)i(i)f(f)f(f)e(e)r(r)e(e)n(n)t(t)i(i)a(a)l(l) e(e)q(q)u(u)a(a)t(t)i(i)o(o)n(n)x′′mak(=)es(√)se(.Fi)nd

Problem 2. (6 marks)

Find the 1-pa¯rameter family of solutions and then solve each of the initial value problems for the following differential equations:

(a) y′ − y = 2te2t,y(0) = 1

Problem 3. (6 marks)

as aSfunc(upp)tion(ose)to(h)f(a)ti(t¯)m(th)e(e) w(iff)here(eren)tw(ia)e(l)as(eq)s(u)u(a)me t(tion)t s(0) .mea(25p)su(−),in(d)mo(escr)nt(ib)h(e)s(s) .a population

(a) Find the time at which the population becomes extinct if p(0) = 750. (b) Find the extinction time if p(0) = p0, where 0 < p0 < 1400.

(c) Find the initial population p0 if the population will be extinct in 2 years.

Problem 4. (4 marks)

Suppose that¯you would like to make five hundred thousand dollars. To do so you plan on investing k dollars per month for 20 years. Assuming that your annual rate of return is six percent compounded continuously, how much must you invest each month?