You can
put this solution on YOUR website! .
= A*e^x - B*sin(x)
is the differential equation you are looking for.
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Side note. This fragment of the post ". . . by eliminating the arbitrary constants" MAKES NO SENSE.
The correct formulation is THIS :
For the given function y = Ae^x + Bcosx, form a differential equation.
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You posted a series of posts with similar / (identical) passage.
I INSISTENTLY recommend you to RE-POST all these problems by EDITING them CORRESPONDINGLY.