Macbook pro update from OS X 10.9.5 (13F1911)

Hard to say what is possible without knowing more about the computer. Older MBPs cannot run the latest OS version, and have different maximum OS levels depending on age.

So, for us to help, we need you to tell us the sub-model, the part with a year and Early, Mid, or Late in the name, from About this Mac. This is the "About" window in OS 10.9 but for an iMac. However, the sub-model will be in the same position as the example:

Also it may help if you tell us what apps (names and versions) say for the OS version they require.

⚠️ NOTE: If the univerity provide the laptop, you may not have the privileges to install a new OS version. Only your IT people, will know.

It depends if you have administrator access to make the required changes. I would also check to make sure you can Option Boot the laptop in case a firmware password has been enabled would make it impossible for you to fix anything if the upgrade goes bad. To Option Boot, hold down the Option key just after the startup chime.

You should definitely check with the university to get permission to perform the upgrade yourself. It is possible the university may have some required software on the laptop that may not work with some newer versions of macOS.

Keep in mind that an OS upgrade is very stressful for a hard drive so you may find that the hard drive will fail during the upgrade so be prepared.

You should make sure you have a backup of all your data before attempting an upgrade.