Final answer:
The solution to the differential equation is given by calculating the roots of its characteristic equation. The roots provide the values for a and β, which are 1 and √5, respectively. These values correspond to the exponential growth/decay rate and frequency of oscillation.
Step-by-step explanation:
The question asks for the values of a and β in the general solution to a second-order differential equation, which is in the form y(x) = eax(c1cosβx + c2sinβx). To solve this, we look at the characteristic equation of the given differential equation d2y/dt2 - 2dy/dt + 6y = 0. The characteristic equation is r2 - 2r + 6 = 0. Solving this quadratic equation gives us complex roots because the discriminant (22 - 4*1*6) is negative, indicating an oscillatory solution. The real part of the roots determines the value of a, and the imaginary part determines the value of β.
The roots are of the form r = a ± βi where i is the imaginary unit. For this particular characteristic equation, using the quadratic formula, we find that a = 1 and β = √5 because the solutions to the characteristic equation are r = 1 ± √5i.