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.

(Mixing tank)For the mixing tank governed by (31):

LetQ(t) =4 for 0 < t < 1 and 2 fort > 1, and letv = c1 = 1and c(0) = 0. Solve forc(t).HINT: The application of (24) in Section 2.2 is not so hard whenq(x)in the differential equationy′+p(x)y = q(x)is defined piecewise (e.g., as in Exercise above), but is tricky when p(x)is defined piecewise. In this exercise we suggest that you use (24) to solve for c(f) first for 0 <t<1, with “a”=0 and “6” = c(0) = 0. Then, use that solution to evaluate c(l) and use (24) again, for 1 <t< ∞ this time with “a”= 1 and “b” = c(l), where c(l) has already been determined.

Exercise

(RL circuit)For theRLcircuit of Example1,suppose that i(0) =i0and thatE(t)is as given below. In each case, determinei(t)and identify the steady-state solution. If a steady state does not exist, then state that. Also, sketch the graph ofi(t)and label key values.