a. If ar^2 + br + c = 0 has equal roots r sub1, show that
L[e^rt] = a(e^rt)’’ + b(e^rt)’ + c = a(r - r sub1)^2 e^rt (i)
b. Differentiate Eq. (i) with respect to r and interchange differentiation with respect to r and with respect to t, thus showing that
(d/dr)L[e^rt ] = L[(d/dr) e^rt ] = L[te^rt ] = ate^rt (r - r sub1)^2 + 2ae^rt (r - r sub1) (ii)
Since the right side of the Eq. (ii) is zero when r = r sub1 conclude that t*exp(r1t) is also a solution of
L[e^rt] = a(e^rt)’’ + b(e^rt)’ + c = a(r - r sub1)^2 e^rt (i)
b. Differentiate Eq. (i) with respect to r and interchange differentiation with respect to r and with respect to t, thus showing that
(d/dr)L[e^rt ] = L[(d/dr) e^rt ] = L[te^rt ] = ate^rt (r - r sub1)^2 + 2ae^rt (r - r sub1) (ii)
Since the right side of the Eq. (ii) is zero when r = r sub1 conclude that t*exp(r1t) is also a solution of
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