Is it permitted to make dua after the obligatory prayers?

Yes it is permitted and is part of the Sunnah. There are several ahadith on the topic:

وعن أبي أمامة رضي الله عنه قال‏:‏ قيل لرسول الله صلى الله عليه وسلم‏:‏ أي الدعاء أسمع‏؟‏ قال‏:‏ ‏ "‏جوف الليل الآخر ودبر الصلوات المكتوبات‏"‏‏

The Messenger of Allah (saw) was asked as to which du’a was most quickly accepted? He replied: ‘In the middle of the night and after the obligatory (fard) prayers’. (Tirmidhi, Hassan)

And the way of making du'a:

"‏ إِذَا دَعَوْتَ اللَّهَ فَادْعُ بِبُطُونِ كَفَّيْكَ وَلاَ تَدْعُ بِظُهُورِهِمَا فَإِذَا فَرَغْتَ فَامْسَحْ بِهِمَا وَجْهَكَ

"When you make du'a to Allah, make du'a with your palms upward, not with the backs of your hands up, and when you finish, wipe your face with them."

The Prophet (saw) made du'a on several occasions in varying circumstances:

1. Between adhan and iqamah
2. Over the dead
3. For those who are absent
4. For those who are present
5. Over food
6. After salah
7. Before sleeping, after waking,
8. For the sick
9. For his ummah
10. For good harvest
11. For rizq
12. For children
13. For rain
14. For changing qadr,
15. Seeing a new moon
16. When seeing your reflection
17. For a birth of a baby
18. To ask Allah to accept another's du'a, and
19. After sacrificing an animal.

Thus it is the sunnah to make du'a on any and every occasion as we would like, as he has not forbidden or restricted this.

لَا يَجْتَمِعُ مَلَأٌ فَيَدْعُو بَعْضُهُمْ، وَيُؤَمِّنُ الْبَعْضُ، إِلَّا أَجَابَهُمُ اللَّهُ

‘No group of believers assemble and they make du’a (together) except that Allah answers their prayers’. (Tabarani & Mustadrak, sahih)