First Order Linear Differential Equations

Yehhh… I know it sounds quite hard, but it’s really not that complicated. They’re equations that come in the form:

$$\frac{dy}{dx}+p(x)y=q(x)$$

To solve this, you’ll need an “integrating factor”. Firstly, the integrating factor is $\mu (x)=e^{\int p(x)dx}$. We do this because we’re next going to multiply everything by it, so we end up with the LHS in the form:

$$\mu (x)\frac{dy}{dx}+\mu (x)p(x)y$$

Note: This looks a lot like the expansion of the product rule (Derivative = $f^{\prime }(x)g(x)+f(x)g^{\prime }(x)$). Thus, you can turn the LHS into $\mu (x)y$.