Ah addition...
if the Mysql authentication is successful you could write down the user password directly in Oracle without the need for chaning
So basicly my app would do something like this
Check Oracle table to see if this user has set password
If yes check login against Oracle
If no then check login again MySQL
IF succesfull write down the password from login to Oracle
if unsuccesfull denied access and offer to reset password in which case reset password would be written in Oracle