NOTE: Thus far we have assumed thatp(x)andq(x)iny′+ p(x)y = q(x)are continuous, yet in app...

NOTE: Thus far we have assumed thatp(x)andq(x)iny′+ p(x)y = q(x)are continuous, yet in applications that may not be the case. In particular, the “input”q(x)may be discontinuous. In Example 1, for instance,E(t)inL di/dt + Ri = E(t)may well be discontinuous, such as

We state that in such cases, whereE(t)has one or more jump discontinuities, the solution (11) [more generally, (24) in Section 2.2] is still valid, and can be used in these exercises.

(Compound interest)Suppose that a sum of money earns interest at a ratek,compounded yearly, monthly, weekly, or even daily. If it is compounded continuously, thendS/dt = kS,where 5(f) denotes the sum at timet.If S(0) = 5o, then the solution is

S(t)=S0ekt. (16.1)

Instead, suppose that interest is compounded yearly. Then aftertyears

S(t)= S0(l + k)t,

and if the compounding is donentimes per year, then

Letk= 0.05 (i.e., 5% interest) and compare S(t)/S0after 1 year (t = 1) if interest is compounded yearly, monthly, weekly, daily, and continuously.