Since the error message includes
y^[Prime][Prime])[x]
that ^ makes me believe you are somehow putting your ' and ' into superscript notation and making Mathematica think you are taking y to the '' power.
If you start Mathematica fresh and just literally type only and nothing more than
A = First[ y /. NDSolve[{y''[x] == x Sin[y[x]], y[0] == 1, y[0] == 1}, y, {x, 0, 19}]];
Plot[A[x], {x, 0, 19}]
without any <ctrl><shift><alt> anything to make anything into a superscript and don't use any palettes to try to publish your notation, just type, or maybe even scrape off the screen and paste the text above, then I think the problem will go away
