2.1 Integration factor

Integrating Factor $\mu (x)$

In general first-order linear differential equation in standard form:

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

In convenient ,we also write the equation in this form:

$$P(t)\frac{dy}{dt} + Q(t) y = G(t)$$

then ,to solve this ODE, we tend to find a integrating factor $\mu (x)$ s.t. $(\mu (t) y)' = k(t)$

How to find the Integrating Factor

$$P(t) \frac{dy}{dt} + Q(t)y = G(t)$$ $$\mu (x) P(t) \frac{dy}{dt} + \mu (x) Q(t)y = \mu (t)G(t)$$ $$\mu (t) \frac{dy}{dt} + \mu (t) \frac {Q(t)}{P(t)}y = \frac {\mu (t) G(t)}{P(t)}$$ $$[\mu (t) y]' = \frac{\mu (t) G(t)}{P(t)}$$

we can imply that $\mu '(t) = \frac{Q(t)}{P(t)} \mu (t)$ ,then

$$\mu (t) = e^{\int_{}^{} \frac{Q(t)}{P(t)} dt}$$