Solution 2.9.6.1 R i(t) Switch closes at t = 0 e(t) L i L (0) Figure 1: Series R;L circuit For the circuit of Figure 1, the governing di erential equation is e(t)=i(t)R+ L di(t) dt : Applying the Laplace transform weobtain E(s) = I(s)R+ L[sI(s);i(0)] This can be rearranged as I(s)= 1 L  E(s);Li L (0) s + R=L  For e(t)=V;; E(s)=V=s: Then for i L (0) = 0, wehave I(s)=  V=L s(s + R=L)  ;; In partial fraction form I(s)= A s + B s + R=L : 1 Then A = sI(s) s=0 =  V=L (s + R=L)  s=0 = V=R B = (s + R=L)I(s) s=;R=L =  V=L s  s=;R=L = ; V R Then i(t)=(V=R)[1;e ;Rt=L ]1(t): 2