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.

(RC circuit)

For theRCcircuit of Example 2, suppose thatio =0 and thatE(t) = E0e−Ri/L.Solve fori(t)and identify the steady-state solution, treating these cases separately:R2C ≠ L,andR2C =L.If there does not exist a steady state, then state that. Sketch the graph of i(t)and label any key values, but withR = C =1 andE(t)= E0sint.