Generally, schools named after a state are public with some exceptions such as Univ. of Penn., New York University, Washington University in StL., which are privates. Schools named after a city, in most part, are privates, i.e., University of San Francisco, Univ. of San Diego, Univ. of Rochester, Univ. of Redlands, Univ. of Richmond, etc. Generally, schools named after a person are private, i.e., Stanford, Vanderbilt, George Washington University, Rice University, Emory University, etc.
Purdue University sounds like a private college, but it is a public college.