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The general solition to the second-order differontal equation dt²d²y​−2dtdy​+6y=0 is in the form y(x)=eax(crcosβx+c2sinβx) find the values of a and β, where β>0. Answer α= and β=

User Meub
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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.

User Imskm
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