Or, is there a way to solve differential equation by symbolic directly?
As already written Mathcad offers no way to solve a DE symbolically but you may try Laplace transformation to get a symbolic result. It would require some manual substitution, though, unless you use Mathcad 11.
http://communities.ptc.com/docs/DOC-1394
http://www.ptc.com/appserver/wcms/resourcecenter/mathcad.jsp?im_dbkey=136068
http://ptc.hosted.jivesoftware.com/docs/DOC-3682
http://electronica.ugr.es/~amroldan/modulos/docencia/pfc/Mathcad/Laplace.pdf
Maybe the true result in the 3rd equation is a coincidence?
No, the "method" of first solving for a constant A and then feeding the (non-constant) result in the equation works for other functions and integral limits as well, as long as the double integral yields a constant, as is the case in your setup. I must confess that I am yet not quite sure why it works ;-) So I would prefer the method Alan had sketched.
