How to solve an order 1 ODEs

Separable form

If a differential equation is of the form $\frac{dy}{dx} = \frac{P(x)}{Q(y)}$ (or equivalently $y’=P(x)Q(y)$), then we solve the equation by \(\int P(x)dx = \int Q(y)dy.\)

Integral factor (non separable form)

If a differential equation is of the form $y’+P(x)y=Q(x)$ (note that the leading coefficient is 1), then we will find its integral factor \(I=e^{\int P(x)dx}\) and the solution is \(y=\int IQ(x)dx.\)