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Please help anyone!..find the particular solution for the given equation and conditions: y"+10y'+29y=0, y=0.5 and y'=1.5 when x=0

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I'm thinking you mean complimentary solution as the particular solution to this differential equation is Y_P(t) = 0 because the differential equation is homogeneous (i.e. the "=0" part). At a quick glance of the differential equation it looks like the following page should help you get the complimentary solution.
I think you can use LaPlace transform to solve this equation.
Thank you that was helpful

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Other answers:

You can use Laplace transforms but a quick characteristic equation (as described in the page above) would probably be just as easy.

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