Differential Equations

1. The Gompertz differential equation dy dt y (a-blny), where y> 0 and a and b are parameters, is used in actuarial studies, and to model growth of objects as diverse as tumors and organizations. (a) Find all equilibrium solutions. (b) Use the substitution z = In y to find the general solution of the Gompertz differential equation. (c) Solve the IVP consisting of the Gompertz differential equation and the initial condition y(0) = yo > 0. (d) Describe the limiting value for y(t) as too. Assume that a > 0, and consider the cases b>0 and b < 0.

2. Find two lincarly independent solutions: ₁(t), ₂(t) of the homogeneous equation: Then show that the so-called Wronskian determinant is not zero. Give an example of two time dependent functions x₁(t), x2(t) for which the Wronskian defined above is zero. Justify your example.

3.5 Determine whether the following models are stable, unstable, or neutrally stable:

3.4 Solve each of the following problems with the trial solution method. Identify the free, forced, transient, and steady-state responses.

3.2 Solve each of the following problems by separation of variables:

The monthly sales of Sunny Electronics' new sound system are given by q(t) - 4,000c - 60² units per month, t months after its introduction. The price Sunny charges is p(t)- 2,000 - ² dollars per sound system, t months after introduction. Find the rate of change of monthly sales. HINT [See Example 4(a).] q'(1) - Find the rate of change of the price. p'(t) = Find the rate of change of monthly revenue 5 months after the introduction of the sound system. $ per month Interpret your answer. When t = the revenue is increasing at a rate of $ per month.

10. If a bank offers 6.5% interest, in how many years will a deposit increase by 50% if the compounding is done: a. Quarterly? b. Continuously?

1) Show that the system described by the equation is linear dy(t) dt - +3y(t) = x(t)

Find dy/dX by implicit differentiation. 5 , 6 , 7 , 8 , 9 , 10

3. Let A and B denote nxn matrices. For each of the following, prove the statement in general or give a counterexample that shows it is not true. (a) (A + B) (A - B) = A²-B² (b) (I+A)² = 1 + 2A + A²