Determine the values of r for which the given differential equation has solutions of the form y = tr for t > 0. (Enter your answers as a

Question

Determine the values of r for which the given differential equation has solutions of the form y = tr for t > 0. (Enter your answers as a comma-separated list.) t2y” + 6ty’ + 6y = 0

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Eva 1 week 2022-01-11T19:25:08+00:00 1 Answer 0 views 0

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    2022-01-11T19:26:35+00:00

    Answer:

    r= [ -2, -3 ]

    Step-by-step explanation:

    for

    y= t^r

    then

    y’  =r*t^(r-1)

    y”= r*(r-1)*t^(r-2)

    thus if

    t²y” + 6t*y’ + 6y = 0

    r*(r-1)*t^(r-2) * t² + 6*t*r*t^(r-1) + 6*t^r = 0

    t^r *[ r(r-1) + 6*r + 6 ] =0

    since t>0 , then [ r(r-1) + 6*r + 6 ] =0

    r(r-1) + 6*r + 6 =0

    r² – r + 6*r + 6 = 0

    r²  + 5*r + 6 = 0

    r₁,r₂ = [-5 ±√(5² – 4*1*6)]/2*1  

    r₁ = (-5+1)/2 = -2

    r₂ = (-5-1)/2 = -3

    then r= [ -2, -3 ]

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