Upgrade 10.4 to 10.5 using disc that shipped with a different MacBook?

I have an older MacBook Pro Intel Core 2 Duo currently running 10.4.11. I need to upgrade to 10.5 Leopard.

A friend gave me the install disc that came with his MacBook (which he no longer uses). It says "Mac OS X 10.5" on it. However, it does not look like the "retail installer" disc (the purple one with the big X). Instead, this disc is gray with white lettering.

Can I upgrade to 10.5 with this disc, or do I really need the "retail installer?" As you may know, those are hard to find. I do not want to do a highly technical install, just a simple upgrade of the OS. I have backed up my files (although I have NOT cloned the HD), but I do not want to proceed with this disc until I'm sure it will work.

I cannot find a clear answer on the forums, so if anyone knows, I would be most grateful. Thanks!