Integrating Factors

So we rewrite the equation as

$$[\mu(t) y]' = \mu(t) g(t)$$

Now integrate to obtain

$$\mu(t) y = \int{\mu(t) g(t)} + c$$

and finally, solve for y

$$y = \frac{\int{\mu(t) g(t)} + c}{\mu(t)}$$

Without an initial condition, we can only obtain a
general solution. General solutions will usually involve an infinite family of solutions, distinguished by different choices for one or more arbitrary constants.

We often refer to the members of this infinite family of solutions as integral curves.

Given an initial condition y(t₀) = y₀, we can single out a unique solution.