There are two ways to get a refund in your situation.
The first is the Lemon Aid Law. If the vehicle failed an inspection within 7 days of the sale, you can cancel the sale within 14 days. You'll need a written statement from the inspection station that it failed and a written statement that the costs of repair exceed 10% of the purchase price. Then you have to notify the seller that you are cancelling the sale by sending them a letter by certified mail. Then physically return the vehicle to the seller, and bring a witness with you so you can confirm it was returned. Once you do all this within 14 days of the sale, you're entitled to a refund. You didn't say if you had the vehicle inspected yet, so if you haven't you should do that. This is the easiest way to get a refund.
The other method is the Used Vehicle Warranty Law. In order to get your money back from a private sale, you'll have to prove that the defect substantially impairs the vehicles use or safety. You'll also need to be able to prove that the seller knew about the defect prior to the sale. If you think you can prove these things, you can cancel the sale any time within 30 days of the sale. Usually a statement from a qualified mechanic can get you this information.
Hope this helps.
Tom is right (he always is) so follow his advise and if it doesn't work out then I recommend you call Tom and see if he can help you. He knows your state laws. Perhaps for a modest fee he can go over all your paperwork (if you have any) and review your situation and help you get it solved. He's right there in your home state. Thanks for asking and good luck.
This answer is for general purposes only and does not establish an attorney-client relationship. The law in your state may differ and your best answer will always come from a local attorney that you meet with privately. If you need a Consumer Law attorney, click the link above to find a Consumer Law attorney near you.