monet A
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Homework Statement
A 2nd order DE with a Delta Dirac forcing function, I have been asked to solve in my DE course.
y'' + 2y' + 10y = \delta(t-1)
Also given are IV's y'(0)=0, y(0)=0
Homework Equations
At this stage in my course I am being asked to only use the method of partial fractions to rewrite my Laplace equations into a recognisable invertable form I am not supposed to be using the integral definition to invert my solutions.
The Attempt at a Solution
The Transforms of each side I have so far obtained:
Y(s).(s^2 + 2s + 10) = e^{-x}
This gives me a solution for Y(s) -->
Y(s) = \frac{ e^{-x} }{s^2 + 2s + 10}
So I have Y(s) = e^{-x} . F(s)
e^{-x} is no problem to invert but F(s) has a quadratic in denominator which cannot be factored. I am at a loss as to how I can find any invertable form for it and for some reason I'm not aware of any other technique that is possible.