How to solve first-order linear ordinary differential equation?

Dy/dx + P(x) = Q(x)

This is the correct initial form.  Convert to this form if necessary.
Next find the integrating factor    ∫ Pdx.
                                                e

Then  Multiply everything by this factor  and verify that left side is product rule.

Integrate both sides.

Solve  for  y

Note sometimes an equation will be exact - that is the left side will be the product rule already so there is no need to employ the integrating factor in this case.