Is it obligatory to recite the Holy Quran with Tajwid (full Tajwid rules and elongations) in the silent prayer or the silent rak'ahs of the audible prayer? And is the prayer valid if this is not done?

Tajwid in prayer and outside of it requires pronouncing the letters from their correct articulation points, without substitution or a linguistic error that changes the meaning. This is obligatory for those who are capable of it. As for other aspects of beautifying the recitation, such as observing the elongations (mudud), it is recommended and not obligatory, and the prayer of one who recites in this manner is valid. Sheikh Ibn Uthaymeen mentioned that tajwid falls under the category of beautifying the voice and is not obligatory.