- #1
Gogeta007
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Homework Statement
I have a couple of problems that I am 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
Homework Equations
y'' = s2Y(s) - sf(0) - f'(0)
y' = sY(s) - f(0)
y = Y(s)
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){s2-6s+9} = (1 +s2)/s2
So I get Y(s) = (1+s2)/s2(s-3)2
IIRC for partial fractions it should be the following: A/(s-3) + B/(s-3)2 + (Cs+D)/s2
I don't 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 can't 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/(s2-6s + 13)
and well. . I am lost from this point onwards. . .I don't remember how to do partial fractions if you can't 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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