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 φ(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 l’hôpital’s rule to evaluate the limit of φ(t; r1, r2) as r2 → r1, thereby obtaining the second solution in the case of equal roots.