In the world of music, the phrases "know by heart" and "play by ear" are pretty peculiar. I would like to offer you some insight on these phrases, being a singer and a piano player myself.
To say "I know it by heart" usually means it's been way more than memorized, it's been internalized. "This song is personal to me," the performer is saying, "I have no problem recalling it. I know it." A performer has probably toiled countless hours practicing and perfecting this song to play and express it the best he can.
We can compare this to getting a testimony of The Book of Mormon. When we read the Book of Mormon with an open heart we can gain knowlege that is ingrained in us.
2 Nephi 33:1 "And now I, Nephi, cannot write all the things which were taught among my people; neither am I mighty in writing, like unto speaking; for when a man speaketh by the power of the Holy Ghost the power of the Holy Ghost carrieth it unto the hearts of the children of men."
The Holy Ghost carries the truth of what we read to our heart. The prophets in the Book of Mormon wrote such things when they were moved upon by the Holy Ghost, just as prophets of old in the Bible. So the Holy Ghost verifies the words they speak to be true. Notice that the Holy Ghost does not carry it into our heart but unto our heart. This takes us to the next phrase.
To Play by Ear is to try to learn a song by listening to the song itself and not having any sheet music to use.
You really have to listen closely to a song and to its very intricate details in order to get the exact same sound. The only way you can get a song to the express uniqueness by which it was written is to find the right combinations of chords and melody. As we listen very closely to the Holy Ghost we can know if the Book of Mormon is true or not. All we need to do is listen and turn down the noise of life so we can actually hear the Holy Ghost testify to our hearts of truth.
1st Kings 19: The Lord speaks to Elijah in a still, small voice.
12 And after the earthquake a fire; but the Lord was not in the fire: and after the fire a still small voice."
Moroni 10:3-5
3 Behold, I would exhort you that when ye shall read these things, if it be wisdom in God that ye should read them, that ye would remember how merciful the Lord hath been unto the children of men, from the creation of Adam even down until the time that ye shall receive these things, and ponder it in your hearts.
4 And when ye shall receive these things, I would exhort you that ye would ask God, the Eternal Father, in the name of Christ, if these things are not true; and if ye shall ask with a sincere heart, with real intent, having faith in Christ, he will manifest the truth of it unto you, by the power of the Holy Ghost.
5 And by the power of the Holy Ghost ye may know the truth of all things.
I encourage you to read the Book of Mormon, to ponder the message it contains. If you read it enough and are prayerful in understanding the teachings, you will come to know for yourself that they testify of Jesus Christ. If you listen to the feelings in your heart and the impressions on your mind and perhaps "read by ear", then you can "know by heart" because the Holy Ghost testified to you the truth.
The Book of Mormon is the Keystone of The Church of Jesus Christ of Latter-Day Saints. If you want to know for yourself, I exhort you to read The Book of Mormon and ask God, with Faith in Christ if it is true believing that God will give you an answer. Be honest in your heart and believing, I believe you will get an answer. God loves all his children and wants to show them the truth of all things if they just ask.