It depends entirely on what fiber your duchess satin is made from. It's easy to dye rayon/silk blend duchess satin, but very difficult to dye polyester duchess satin.

Dylon makes several different kinds of dye that will work well on the rayon/silk blend type of duchess satin. Both rayon and silk are easily dyed, using Dylon Hand Dye, Dylon Machine Dye, Dylon Cold Dye, Dylon Permanent Dye, or even Dylon Multi Purpose dye; the last one listed is the most prone to fading. Dylon Cold Dye can be used in lukewarm water, and therefore is suitable for duchess satin that is labeled "wash in cold water", though you must know that there is no dye that can be safely used if your fabric is labeled "dry clean". (You can't dye anything that is not washable.)

There is no Dylon dye that will work on polyester satin. The only dye that works on polyester fabric is called disperse dye. You can buy disperse dye by mail-order only. Fibrecrafts in the UK and Blick Art Materials in the US sell "iDye Poly"; PRO Chemical & Dye in the US sells "PROsperse" direct dye; Batik Oetoro in Australia sells "PolySol" disperse dye. None of these can be used without extensive boiling; you cannot use them on your polyester satin if it is labeled "dry clean only" or even "wash in cold water".

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[This answer was first posted, by me, on Yahoo answers, on October 11, 2008.]