See Boas 6.8 and 8.4

Note: Any first order linear ODE can be solved on some interval. You can always multiply the differential form of the differential equation by an appropriate function of the independent and dependent variables so that the equation becomes exact. Then you can solve the differential equation using methods for exact equations. Boas 8.3 has a nice description that not only shows you how to find the integrating factor, but also shows you how to find the solution of the differential equation in terms of that integrating factor.