Integrating factor

The integrating factor is a method used to solve first-order linear differential equations, which are in the form:

$$\frac{dy}{dt}+p(t)y(t)=g(t)$$

Generally this method tends to be more reliable than separation of variables. Our IF is:

$$e^{\int p(t)dt}$$

We take the simplest antiderivative in the exponent, where $C=0$. We multiply both terms of the ODE by the integrating factor, then simplify the side with the derivative so that we express it as an explicit derivative of some function, then integrate.

So our problem solving method is:

$$\frac{dy}{dt}y(t)e^{\int p(t)dt}+p(t)y(t)e^{\int p(t)dt}=y(t)e^{\int p(t)dt}$$

which simplifies to:

$$\frac{d}{dt}(y{e}^{\int p(t)dt})=g(t)e^{\int p(t)dt}$$

which is consistent with how we understand implicit differentiation.