Bush did it the right way, the constitutional way. He got Congress to vote approval of his actions the whole step of the way. Those votes were a de jure declaration of war. The Constitution gives the power of declaring war to Congress and Congress voted twice, once in 2001 and later in 2002. Remember, a declaration of war does not have to be headed Declaration of War. It only has to give the authorization, and the ones in 2001 and 2002 did.
Why the de jure work around instead of a Constitutional formal declaration of war? Bush did not measure up nor has Obama.