If you have soldiers who lived in Michigan in your family tree it's possible they ended up in the Old Soldier's Home in Grand Rapids at one point or another. So far, I have found three people from my tree in the database. Some soldiers died in the home and some resided there for a while and then left. One of my soldiers was in and out of the home several times. Widows of soldiers could also apply for admission. It was not necessary that a soldier served from the state of Michigan, only that he lived in the state.
It's easy to determine if one of your people spent time in the Soldier's Home with the free Veterans database provided by the West Michigan Genealogical Society (WMGS). You can also check the Find A Grave site to see if one of your men was buried in the Soldier's Home cemetery.
I first became aware of the Soldier's Home through a newspaper account. My gg-grandmother's brother, Solon Lane, walked to Kalamazoo from Van Buren county after quarreling with his girlfriend. His sister wouldn't admit him to her home so the proprietor of the Columbia House took him in for the night. Lane said he would walk to Hastings where he had friends.  Upon reaching Hastings, Solon Lane appeared before the probate judge bearing his honorable discharge certificate from his Civil War service and wearing his tattered army jacket. He asked to be sent to the Soldier's Home. The papers were summarily filled out and Lane was put aboard a train for Grand Rapids. He reportedly said “that at last his request to be 'buried alongside the old boys' when he died would be granted.” 
If you are fortunate enough to find one of your people in the Soldier's Home database you can order their records with a few clicks. The search results screen indicates how many pages long the file is and the price (ranging from about a $1/page for short files (5-7 pages) up to about $0.60/page for longer files (about 30 pages)). A WMGS member will copy the record and send it to you.
I requested the file for Solon Lane to see what I could learn about him. I already knew quite a bit about Solon from his Civil War Pension application file (Why EveryoneShould Use Military Pension Files), for instance, that he was an unapologetic bigamist having married four women without ever obtaining a divorce. But I digress. The papers in Solon's file (12 pages) consisted of his initial application for admission to the home as well as several applications for re-admission. The re-admission pages had little more information than the dates of admission and discharge. The initial application had a bit more information, including date/place of enlistment and discharge and the unit in which he served, place of birth, occupation, marital status, physical description, any disabilities and whether the person could read and write. In addition, if the person was receiving a military pension it provided the certificate number and if the soldier hadn't served with a Michigan military unit, how long he had resided in the state. Depending on what you already know you may learn something new, or at least be able to piece together a bit more of your person's whereabouts.
As far as I am aware, the Grand Rapids facility was the only Soldier's Home located in Michigan, but some states had more than one. Even if your man wasn't in the Michigan Soldier's home, it is worth checking a database at Ancestry.com, U.S. National Homes for Disabled Volunteer Soldiers, 1866-1938.
This database includes information cards for twelve homes over various years. I found two of my people here, one was Solon Lane and another was one of my veterans who I hadn't realized ever lived in Illinois (and yes, I'm sure it's him).
- “Aged Man Barred From Home In City,” Kalamazoo [Kalamazoo, Michigan] Gazette, 22 December 1908, page 1, column 4, digital images, GenealogyBank (http://www.genealogybank.com/gbnk/: accessed 5 September 2011), Kalamazoo Gazette Collection (Newspaper Archives).
- “Old Soldier Sure Of Home For Life Time,” Kalamazoo [Kalamazoo, Michigan] Evening Telegraph, 28 December 1908, page 10, column 4, digital images, Kalamazoo Public Library (http://www.kpl.gov: accessed 26 June 2012), Kalamazoo Telegraph Collection.