Why does this FO linear DE need an integrating factor?

Question

This guy solves

$$(t+y+1)\textrm{d}t - \textrm{d}y = 0$$

using an integrating factor $\mu(t)=e^{-t}$ and obtains

$$y=-t-2+Ce^{-t}$$

The way I did it was more naive:

$$\int \textrm{d}y = \int (t + y + 1) \textrm{d}t$$

Why is my method incorrect?

Answer 1

Your way is not incorrect, just that on the right you're back to square one, since you have the integral $\int y\mathrm dt$ to evaluate, whereas you need to know the relationship between $y$ and $t$ to do this, and that's what you wanted originally.