Hi mlmcholm,
I've had a good search in ref: to David Burns born 17th Jan 1814. and his Father Francis. I have found a Mary Jardine, as you mention, same town i.e. Applegarth, Sibbaldbie. but rule her out as she was producing her own Daughter, Mary in 1814.
Another Jardine I found was a Janet Jardine, same Town as above, but married to a Francis Cowan.
Sadly, I could not put Francis Burns with any name Jardine ,except at the Birth of David, in a 30 years life-span either way, even Death Records for him do not show up, or Birth details.
I presume you are doublie sure the facts you have are spot on, otherwise I could search in another direction.........................Ron.