Create a textplot showing only the labels, and then merge this with your main plot using plots[ display ]. You might need some trial and error to get the best coordinates for the label locations.

P := plot( [ x^2 , x^3 ] ) :
L := plots[textplot]( [ [  7 , 120 , 'x^2' ] ,  [  5 , 210 , 'x^3' ] ] , axes = none) :
plots[display]( P , L , axes = normal )

If you set

printlevel := 30 ;

and run your command again, you will see what the problem is. Maple solves v'(x) = 0 to obtain v = c (a constant). It then solves the second ODE, i.e. u''(x) - c u'(x) = 1, but the solution doesn't make sense when c = 0. This has to be treated as a special case.