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Unformatted text preview: 2 MA 36600 LECTURE NOTES: FRIDAY, JANUARY 23 The rate of change of the quantity of salt in the solution is the difference of the rate which flows in and the rate which flows out. Hence our initial value problem is dQ 1 Q(t) = ·r− · r, Q(0) = Q0 . dt 4 100 One method of finding QL is to solve the initial value problem for Q(t), then consider the behavior as t increases without bound. We can solve the differential equation using an integrating factor. Note that this equation is a first order linear equation: r r r r dQ + Q= =⇒ p(t) = , g (t) = . dt 100 4 100 4 The integrating factor is ￿￿ t ￿ ￿r ￿ µ(t) = exp p(τ ) dτ = exp t = ert/100 . 100 Multiplying the differential equation above by this function yields the equation ￿r dQ d ￿ rt/100 µ(t) + µ(t) p(t) Q = µ(t) g (t) =⇒ e Q(t) = ert/100 . dt dt 4 Integrating both sides gives ert/100 Q(t) = 25 ert/100 + C =⇒ Q(t) = 25 + C e−rt/100 . =⇒ QL = lim Q(t) = 25 lbs. For the initial condition, set t = 0 to find C = Q0 − 25. Hence the solution to the initial value problem is Q(t) = 25 + (Q0 − 25) e−rt/100 t→∞ We give a second method of finding QL . Consider the differential equation Q￿ + p(t) Q = g (t), and take the limit as t increases without bound: QL ￿ + p(t) QL = g (t). Since QL is a constant, its derivative is zero. Hence g (t) r/4 0 + p(t) QL = g (t) =⇒ QL = = = 25 lbs. p(t) r/100 This trick works in many cases: Consider the first order differential equation dy = G(t, y ). dt Say that we wish to find an equilibrium solution yL = lim y (t). t→∞ If we take the limit as t increases without bound, we find that yL ￿ = G(t, yL ). Since yL is a constant, we actually seek solutions yL to the equation G(t, yL ) = 0 which hold for all time t. Interest Rate Problems. Say that you wish to invest $1000 for the next year. You have two banks to choose from: Bank A will give interest compounded quarterly, whereas Bank B will give interest compounded daily. The rate of interest is determined by the Federal Reserve, say at r = 8% APR. Which bank gives you the better return on your investment? We make sense of these terms. Let S (t) denote the sum of money in the bank at time t. Note that S (0) = $1000. If this interest rate r is compounded m times per year then the rate is r/m for that one round, and so we have ￿ r ￿mτ S (t + τ ) = 1 + S (t) m after a time period τ years elapses. We wish to consider τ = 1 year. Note that m = 4 for interest compounded quarterly, and m = 365 for interest compounded daily. In fact, ￿ r ￿m lim 1 + = er m→∞ m so that we may approximate daily compounding with continuous compounding i.e., m = ∞. (In particular, ￿ r ￿mt dS S (t) = 1 + S (0) ≈ ert S (0) =⇒ ≈ r S. m dt This means we can approximate interest problems with first order linear differential equations.) Consider the ratio: ...
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