In the Aranya Kanda of the Ramayana, when Rama meets the great bird Jatayu for the first time, this is how Jatayu introduces himself:
But that eagle with sweet and soft words spoke to them as though to please them, "oh boy Rama know me as your father's friend." Accepting that bird as his father's friend Raghava revered it... Raghava revered Jatayu in his turn by gladly hugging and stood by him with his head bent, and that kind-natured Rama is indeed regardful of the friendship of his father with Jatayu that is repeatedly said by Jatayu.
My question is, how exactly did Jatayu and Rama's father Dasharatha become friends according to Hindu scripture? After all, it's not every day you make friends with a giant bird! The passage I quoted above seems to suggest that Jatayu told Rama exactly how he knew his father, but unfortunately Valmiki didn't see fit to tell us.
There are three potential stories I've heard involving Dasharatha and Jatayu, but I'm not sure which of them has a scriptural basis:
There is the story told in this webpage about how when Ravana asks Brahma for immortality, Brahma refuses, saying that he is destined to be killed by the son of Kausalya. This takes place before Dasharatha and Kausalya are married. To prevent Kausalya from having a son, Ravana kidnaps her and puts her in a wooden box, which he sets afloat in the Sarayu river. Dasharatha sees the box and jumps into the river to retrieve it. He gets swept away in the current as well, but then Jatayu sees him and rescues him. Dasharatha and Jatayu then team up to retrieve the wooden box. They find Kausalya inside it, Dasharatha and Kausalya get married, and Dasharatha and Jatayu become friends.
In the Kamba Ramayana (the Tamil retelling of the Ramayana), when Jatayu hears from Rama that Dasharatha is dead, he laments the loss of his friend, saying this:
2712.Oh Dasaratha who has a pretty mind without any guilt , When you like the churner which breaks the curd killed Sambarasura , Who was giving sorrow to the beings of the world , you said , Making the world scattered with soft dust and surrounded by the sea that, “You are the body and I am the soul”, and your words will not change , But with soul lying here , the ignorant Yama has taken the body to the heaven.
Now Sambarasura, AKA Timidhwaja, is a demon that is mentioned in the Ayodhya Kanda of the Ramayana. He lived in the Dandaka forest, and was such a formidable demon that the gods enlisted the help of Dasharatha in defeating him. After the battle Dasharatha was badly wounded, but his wife Kaikeyi drove the chariot and brought him back to safety. Dasharatha was so grateful to Kaikeyi for saving his life that he gave her the infamous two boons which were later used to exile Rama to the forest. The Ayodhya Kanda doesn't mention Jatayu, but it's possible that Jatayu fought alongside Dasharatha in his battle against Sambarasura.
There is the story told in this excerpt from the new Amar Chitra Katha comic "Divine Beings". Once, during Dasharatha's early days as king of Ayodhya, the kingdom suffered a terrible drought. Dasharatha consulted his priests, who informed him that the kingdom was suffering the ill-effects of Shani (god of the planet Saturn). So Dasharatha took off in a flying chariot to find Shani and persuade him to have mercy on Ayodhya. When Shani saw Dasharatha approaching, he feared that his gaze might kill Dasharatha, so he instead directed his gaze at Dasharatha's chariot, which was burnt to ashes. Dasharatha started falling to the Earth, but he was caught by Jatayu, who had been watching the whole thing. In any case, Shani was pleased by all the efforts Dasharatha had gone through, so he agreed to stop Ayodhya's drought. Jatayu flew Dasharatha back to Ayodhya, and they were friends ever since.
So which of these three stories, if any, has a basis in Hindu scripture?
I'm inclined to doubt the veracity of the first story, because in the Yuddha Kanda of the Ramayana, Ravana quotes Brahma as just saying "You know of the threat from men", not indicating which human it was going to be. I'm inclined to believe that the story in the Kamba Ramayana is probably true as far as it goes, because the author Kamaban was very well-researched, but Kamban never claims that this is how Jatayu and Dasharatha first met. So maybe Jatayu fought alongside Dasharatha after they had already become friends. So that leaves the story given in the Amar Chitra Katha comic, which to my mind seems like the most plausible account of how Jatayu and Dasharatha first met.