A little confused in Zero Input response

AI Thread Summary
The discussion centers on finding the zero input response for a differential equation involving complex roots. The equation presented is (D-1)(D^2+1)y(t)=(D^2+2)f(t), with initial conditions y(0)=4, y'(0)=3, and y''(0)=3. The user is familiar with simpler cases but struggles with the presence of complex roots, specifically involving the imaginary unit j. They seek a step-by-step explanation to better understand how to incorporate these complex roots into their solution. Clarification on handling the j's and their identities is requested to resolve the confusion.
zee3b
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Guys I know how to solve the simple ones but this one has complex roots
I have to find the zero input response

(D-1)(D2+1)y(t)=(D2+2)f(t)

y(0)=4 y'(0)=3 y''(0)=3

D2= D square. Y' = first derivative Y''= secondI know the roots are +1, j and -j

But I don't know how to solve or use the j's and their identity

I maybe wrong. If it wasn't for the square inside the parenthesis I would be able to solve it.

Any help would be appreciated. (step by step so i can understand it better) once again I know the whole concept but not the j's
 
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Any one? :(
 
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