Differntial Equations

First Order Linear Differential Equation

A first order linear differntial equation has the following form:

$$ \frac{dy}{dx} +P\left( x \right) y=Q\left( x \right) $$

Integrating Factor

$$ \frac{dy}{dx} +P\left( x \right) y=Q\left( x \right) $$

multiyplying the expression by the integrating factor $\mu(x)$

$$\mu(x) \frac{dy}{dx} + \mu(x) P\left( x \right) y= \mu(x) Q\left( x \right) $$

Setting: $\mu^{\prime}(x) := \mu(x) P\left( x \right)$ and finding the integrating factor:

$$\mu^{\prime}(x) = \mu(x) P\left( x \right)$$ $$ \Leftrightarrow \frac{\mu^{\prime}(x)}{\mu(x)} = P\left( x \right) $$ $$ \Leftrightarrow \int \frac{\mu^{\prime}(x)}{\mu(x)} dx = \int P\left( x \right) dx $$ $$ \Leftrightarrow \ln \left( \left| \mu(x) \right| \right) = \int P\left( x \right) dx $$ $$ \Rightarrow \mu(x) = e^{ \int P\left( x \right) dx} $$