question about solving dfq's, PLEASE RESPOND

Hey all, i just lost a TON of points on a test for solving a problem in a way that is apparently invalid.

The problem was verify that y(x) = x+1 is a solution for dy/dx = y*y-x*x-2x; y(0) = 1. i plugged y = x+1 into the right side of the second equation, got dy/dx = 1, integrated to get y=x+c, used y(0) = 1 to get c= 1, therefore y = x + 1

My professor's (Angry) annoyed 2-second explanation about why my method is invalid was that I assumed that it worked to prove that it worked. I sorta buy it, but I'm not completely convinced, could someone give me a counter example to prove that my method is not legit? [to clarify, my method is to plug in y(x) into the DE, then integrate, then use the given initial conditions to solve for c to get a new y(x) and make sure that my new y(x) is the same as the old one].

The counter example I am requesting would take a form that is similar to the problem above, except that y(x) would not be a legit solution to dy/dx, BUT my method would falsely show that y(x) does work. Obviously, if no such counter example exists, that my method proves that the DE works and I should not have lost any points

Thanks in advance! (Happy)