The state does not need to prove that there was a motive in a homicide prosecution for murder.
A motive is a reason to do something. It is the driving force behind an action. You eat because you are motivated by your hunger. You drive because you are motivated by your desire to go somewhere.
However, in a prosecution for murder, the state is not required to prove that there was a motive to kill on behalf of the accused.
In a first-degree murder prosecution, the charge requires the state to prove premeditation, or planning. That, in and of itself, does not necessarily equate to a motive. Somebody could plan to kill somebody just for the heck of it.
In a second-degree murder prosecution, the state need not prove premeditation or motive as well. All the state needs to prove is that somebody was killed due to the accused's depraved act.
It is a misconception that the state needs to have proof of a motive in order to establish a successful homicide prosecution. A motive may help jurors to understand the evidence better but it is not an element of the crime that must be proven by the prosecutor beyond a reasonable doubt.