20 through 22 indicate other ways of finding the second
Chapter 3, Problem 21(choose chapter or problem)
20 through 22 indicate other ways of finding the second solution when the characteristic equation has repeated roots.Suppose that r1 and r2 are roots of ar2 + br + c = 0 and that r1 = r2; then exp(r1t)and exp(r2t) are solutions of the differential equation ay+ by+ cy = 0. Show that(t;r1,r2) = [exp(r2t) exp(r1t)]/(r2 r1) is also a solution of the equation for r2 = r1.Then think of r1 as fixed, and use LHpitals rule to evaluate the limit of (t;r1,r2) asr2 r1, thereby obtaining the second solution in the case of equal roots.
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