Can you describe the austerity undertaken by the Deshmukhin to get a mantra from Baba?

Certainly. As recounted in Chapter 18, an elderly woman named Radhabai, the Deshmukhin, was determined to receive a secret mantra directly from Sai Baba. To compel him to grant her wish, she made a firm resolve and sat in a 'dharna,' which is a form of fasting protest. Her austerity was extreme; she gave up eating and drinking entirely, not even taking a sip of water. This went on for three days and nights, and she was prepared to fast unto death, a practice known as 'Prayopavesha'. Her penance was so severe that she became exhausted and despondent, causing great worry for others like Madhavrao.