(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

I have a couple of problems that im stuck on. The following:

y'' - 6y' + 9y = t

y(0) = 0

y'(0) = 1

and

y'' - 6y' + 13y = 0

y(0) = 0

y'(0) = -3

2. Relevant equations

y'' = s^{2}Y(s) - sf(0) - f'(0)

y' = sY(s) - f(0)

y = Y(s)

3. The attempt at a solution

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for the first one:

once I substitute into the original equation, I can move things around and I came out with the following:

Y(s){s^{2}-6s+9} = (1 +s^{2})/s^{2}

So I get Y(s) = (1+s^{2})/s^{2}(s-3)^{2}

IIRC for partial fractions it should be the following: A/(s-3) + B/(s-3)^{2}+ (Cs+D)/s^{2}

I dont know if this is where i messed up but I got:

A=-2/27

B=-1/9

C=2/27

D=1/9

as a final result I get:

-2/27e^3t - 1/92e^3t and the other part (Cs + D)/s^2 . . .I cant find any way to transform that

the back of the book says:

1/9t + 2/27 - 2/27e^3t + 10/9te^3t

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The second one:

Starting by substitution, plugging in the values and solving for Y(s)

Y(s) = -3/(s^{2}-6s + 13)

and well. . Im lost from this point onwards. . .I dont remember how to do partial fractions if you cant factorize that denominator. And the quadratic formula (A=1 B = 6 C=13) gives imaginary numbers

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Thank you!

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# Homework Help: Differential Eq: Inverse Laplace Transforms

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