Shouldn't you remove the integrating factor?

One method for doing an ODE is to try to make it an exact ODE by multiplying by an integrating factor. Why don't you have to remove the integrating factor afterwards?

For instance, in this example, the problem was not an exact DiffEQ:

, so NOT an exact DiffEQ.

Next is to look for an integrating factor, which we call :

So, back to the top, multiply the whole equation by x:

(some constant)

**Why don't we have to compensate for after this last step?** Notice the and do not equal the starting and :

And dividing psi by x also doesn't work (although dividing the partial derivatives of psi by x works). What gives here?

Thanks (sorry for the too long post),