Integrating Factor

You can save some trouble with computation if you understand the purpose of an integrating factor.

IMPORTANT: Before computing the integrating factor, put the first-order linear equation into the form $y^\prime + P(x) y = Q(X)$.

When you multiply by the integrating factor $I(x)$, you get $$I(x)y^\prime+I(x)P(x)y=I(x)Q(X)$$.

By design of $I(x)$, this becomes $$\left[I(x)y\right]^\prime =I(x)Q(x)$$.

Side Remark: Keep in mind that $y$ is a function of $x$ when differentiating.

Another side remark: Remember that $y^\prime$ is notation for $\frac{dy}{dx}$.