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) =constant =Qand c(0) = Co. Solve forc(t).
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